SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case,...

8
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 650870, 7 pages http://dx.doi.org/10.1155/2013/650870 Review Article A Priori Bounds in and in 2, for Solutions of Elliptic Equations Sara Monsurrò and Maria Transirico Dipartimento di Matematica, Universit` a di Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy Correspondence should be addressed to Sara Monsurr` o; [email protected] Received 21 March 2013; Accepted 15 May 2013 Academic Editor: Rodrigo Lopez Pouso Copyright © 2013 S. Monsurr` o and M. Transirico. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. e main theorem consists in an -a priori bound, >1. Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given. 1. Introduction e aim of this work is to give an overview on some recent results dealing with the study of a certain kind of the Dirichlet problem in the framework of unbounded domains. To be more precise, given an unbounded open subset Ω of R , ≥2, we are concerned with the elliptic second-order linear differential operator in variational form =− ∑ ,=1 ( + )+ ∑ =1 + , (1) with coefficients ∈ ∞ (Ω) and with the associated Dirich- let problem ∈ ∘ 1,2 (Ω) , = , ∈ −1,2 (Ω) . (2) As far as we know, were Bottaro and Marina the first to approach this kind of problem who proved, in [1], an existence and uniqueness theorem for the solution of problem (2), for ≥3, assuming that ∈ ∞ (Ω) , , = 1, . . . , , (3) , ∈ (Ω) , = 1, . . . , , ∈ /2 (Ω) + ∞ (Ω) , (4) − ∑ =1 ( ) ≥ , ∈ R + . (5) e study was later on generalized in [2] weakening the hypothesis (4) by considering coefficients , , and satisfy- ing (4) only locally and for ≥2. Further improvements have been achieved in [3], for ≥3, since the , , and are taken in suitable Morrey type spaces with lower summabilities. In [1–3], the authors also provide the bound ‖‖ 1,2 (Ω) ≤ −1,2 (Ω) , (6) giving explicit description of the dependence of the constant on the data of the problem. In two recent works, [4, 5], considering a more regular set Ω and supposing that the lower order terms coefficients are as in [3] for ≥3 and as in [2] for =2, we prove that if ∈ 2 (Ω) ∩ ∞ (Ω), then there exists a constant , whose dependence is completely described, such that ‖‖ (Ω) ≤ (Ω) , (7)

Transcript of SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case,...

Page 1: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 650870 7 pageshttpdxdoiorg1011552013650870

Review ArticleA Priori Bounds in 119871119901 and in1198822119901 for Solutions ofElliptic Equations

Sara Monsurrograve and Maria Transirico

Dipartimento di Matematica Universita di Salerno Via Ponte Don Melillo 84084 Fisciano Italy

Correspondence should be addressed to Sara Monsurro smonsurrounisait

Received 21 March 2013 Accepted 15 May 2013

Academic Editor Rodrigo Lopez Pouso

Copyright copy 2013 S Monsurro and M Transirico This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partialdifferential equations in divergence form and with discontinuous coefficients in unbounded domains The main theorem consistsin an 119871119901-a priori bound 119875 gt 1 Some applications of this bound in the framework of non-variational problems in a weighted anda non-weighted case are also given

1 Introduction

The aim of this work is to give an overview on some recentresults dealingwith the study of a certain kind of theDirichletproblem in the framework of unbounded domains To bemore precise given an unbounded open subset Ω of R119899119899 ge 2 we are concerned with the elliptic second-order lineardifferential operator in variational form

119871 = minus

119899

sum

119894119895=1

120597

120597119909119895(119886119894119895

120597

120597119909119894+ 119889119895) +

119899

sum

119894=1

119887119894120597

120597119909119894+ 119888 (1)

with coefficients 119886119894119895 isin 119871infin(Ω) and with the associated Dirich-

let problem

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 119882minus12

(Ω)

(2)

As far as we know were Bottaro and Marina the firstto approach this kind of problem who proved in [1] anexistence anduniqueness theorem for the solution of problem(2) for 119899 ge 3 assuming that

119886119894119895 isin 119871infin(Ω) 119894 119895 = 1 119899 (3)

119887119894 119889119894 isin 119871119899(Ω) 119894 = 1 119899

119888 isin 1198711198992

(Ω) + 119871infin(Ω)

(4)

119888 minus

119899

sum

119894=1

(119889119894)119909119894ge 120583 120583 isin R+ (5)

The study was later on generalized in [2] weakening thehypothesis (4) by considering coefficients 119887119894 119889119894 and 119888 satisfy-ing (4) only locally and for 119899 ge 2 Further improvements havebeen achieved in [3] for 119899 ge 3 since the 119887119894 119889119894 and 119888 are takenin suitable Morrey type spaces with lower summabilities

In [1ndash3] the authors also provide the bound

11990611988212(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817119882minus12(Ω) (6)

giving explicit description of the dependence of the constant119862 on the data of the problem

In two recent works [4 5] considering a more regularset Ω and supposing that the lower order terms coefficientsare as in [3] for 119899 ge 3 and as in [2] for 119899 = 2 we prove thatif 119891 isin 119871

2(Ω) cap 119871

infin(Ω) then there exists a constant 119862 whose

dependence is completely described such that

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (7)

2 Abstract and Applied Analysis

for any bounded solution 119906 of (2) and for every 119901 gt 2This can be done taking into account two different signhypotheses namely (5) and the less common

119888 minus

119899

sum

119894=1

(119887119894)119909119894ge 120583 120583 isin R+ (8)

Successively in [6] we deepen the study begun in [4 5]showing that to a bounded datum 119891 isin 119871

2(Ω) it corresponds

a bounded solution 119906 This allows us to prove by means of anapproximation argument that if 119891 belongs to 1198712(Ω) cap 119871119901(Ω)119901 gt 2 then the solution is in 119871

119901(Ω) too and verifies (7)

Putting together the two preliminary 119871119901-estimates 119901 gt 2obtained under the different sign assumptions and adding thefurther hypothesis that the 119886119894119895 are also symmetric by meansof a duality argument we finally obtain (7) for 119901 gt 1 for eachsign hypothesis assuming no boundedness of the solutionand for 119891 isin 119871

2(Ω) cap 119871

119901(Ω)

To conclude we provide two applications of our final119871119901-bound 119901 gt 1 recalling the results of [7 8] where our

estimate plays a fundamental role in the study of certainweighted and non-weighted non-variational problems withleading coefficients satisfying hypotheses of Mirandarsquos type(see [9])Thenodal point in this analysis is the existence of thederivatives of the leading coefficients that allows us to rewritethe involved operator in variational form and avail ourselvesof the above-mentioned a priori bound

Always in the framework of unbounded domains thestudy of different variational problems can be found in[10 11] Quasilinear elliptic equations with quadratic growthhave been considered in [12] In [13ndash15] a very generalweighted case with principal coefficients having vanishingmean oscillation has been taken into account

2 A Class of Spaces of Morrey Type

In this section we recall the definitions and the main prop-erties of a certain class of spaces of Morrey type where thecoefficients of our operators belong These spaces generalizethe classical notion of Morrey spaces to unbounded domainsand were introduced for the first time in [3] see also [16]for some details Thus from now on let Ω be an unboundedopen subset of R119899 119899 ge 2 By Σ(Ω) we denote the 120590-algebraof all Lebesgue measurable subsets of Ω For 119864 isin Σ(Ω) 120594

119864

is its characteristic function |119864| its Lebesgue measure and119864(119909 119903) = 119864 cap 119861(119909 119903) (119909 isin R119899 119903 isin R+) where 119861(119909 119903) is theopen ball with center in 119909 and radius 119903 The class ofrestrictions to Ω of functions 120577 isin 119862

infin

∘(R119899) is D(Ω) For

119902 isin [1 +infin[ 119871119902loc (Ω) is the class of all functions 119892 Ω rarr R

such that 120577 119892 isin 119871119902(Ω) for any 120577 isin D(Ω)For 119902 isin [1 +infin[ and 120582 isin [0 119899[ the space of Morrey type

119872119902120582(Ω) is made up of all the functions 119892 in 119871119902loc (Ω) such that

10038171003817100381710038171198921003817100381710038171003817119872119902120582(Ω) = sup

120591isin]01]

119909isinΩ

120591minus1205821199021003817100381710038171003817119892

1003817100381710038171003817 119871119902(Ω(119909120591)) lt +infin (9)

equipped with the norm defined in (9)

The closures of 119862infin∘(Ω) and 119871infin(Ω) in119872119902120582(Ω) are deno-

ted by119872119902120582∘ (Ω) and 119902120582(Ω) respectivelyThe following inclusion holds true

119872119902120582

∘(Ω) sub

119902120582(Ω) (10)

Moreover

119872119902120582

(Ω) sube 11987211990201205820 (Ω) if 1199020 le 119902

1205820 minus 119899

1199020le120582 minus 119899

119902

(11)

We put 119872119902(Ω) = 1198721199020(Ω) 119902(Ω) =

1199020(Ω) and

119872119902

∘ (Ω) = 1198721199020

∘ (Ω)Now let us define the moduli of continuity of functions

belonging to 119902120582(Ω) or 119872119902120582∘ (Ω) For ℎ isin R+ and 119892 isin

119872119902120582(Ω) we set

119865 [119892] (ℎ) = sup119864isinΣ(Ω)

sup119909isinΩ|119864(1199091)|le1ℎ

10038171003817100381710038171198921205941198641003817100381710038171003817119872119902120582(Ω) (12)

Given a function 119892 isin 119872119902120582(Ω) the following characteriza-

tions hold

119892 isin 119902120582

(Ω) lArrrArr limℎrarr+infin

119865 [119892] (ℎ) = 0

119892 isin 119872119902120582

∘(Ω)

lArrrArr limℎrarr+infin

(119865 [119892] (ℎ) +1003817100381710038171003817 (1 minus 120577ℎ) 119892

1003817100381710038171003817119872119902120582(Ω)) = 0

(13)

where 120577ℎ denotes a function of class 119862infin119900(119877119899) such that

0 le 120577ℎ le 1

120577ℎ|119861(0ℎ)

= 1

supp 120577ℎ sub 119861 (0 2ℎ)

(14)

Thus if 119892 is a function in 119902120582(Ω) amodulus of continuity of119892 in 119902120582(Ω) is a map 119902120582[119892] R+ rarr R+ such that

119865 [119892] (ℎ) le 119902120582

[119892] (ℎ)

limℎrarr+infin

119902120582

[119892] (ℎ) = 0(15)

While if 119892 belongs to119872119902120582119900(Ω) amodulus of continuity of 119892 in

119872119902120582

119900(Ω) is an application 120590119902120582

119900[119892] R+ rarr R+ such that

119865 [119892] (ℎ) +1003817100381710038171003817 (1 minus 120577ℎ) 119892

1003817100381710038171003817119872119902120582(Ω) le 120590119902120582

119900[119892] (ℎ)

limℎrarr+infin

120590119902120582

119900[119892] (ℎ) = 0

(16)

We finally recall two results of [4 7] obtained adapting toour needs a more general theorem proved in [17] providingthe boundedness and some embedding estimates for themultiplication operator

119906 997888rarr 119892119906 (17)

where the function 119892 belongs to suitable spaces of Morreytype

Abstract and Applied Analysis 3

Theorem 1 If 119892 isin 119872119902120582(Ω) with 119902 gt 2 and 120582 = 0 if 119899 = 2

and 119902 isin ]2 119899] and 120582 = 119899 minus 119902 if 119899 gt 2 then the operator in(17) is bounded from

∘

11988212(Ω) to 1198712(Ω) Moreover there exists

a constant 119862 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 1198712(Ω) le 119862

10038171003817100381710038171198921003817100381710038171003817119872119902120582(Ω)11990611988212(Ω) forall119906 isin

∘

11988212(Ω) (18)

with 119862 = 119862(119899 119902)Let 119901 gt 1 and 119903 119905 isin [119901 +infin[ If Ω is an open subset of R119899

having the cone property and 119892 isin 119872119903(Ω) with 119903 gt 119901 if 119901 = 119899

then the operator in (17) is bounded from 1198821119901(Ω) to 119871119901(Ω)

Moreover there exists a constant 119888 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 119871119901(Ω) le 119888

10038171003817100381710038171198921003817100381710038171003817119872119903(Ω)1199061198821119901(Ω) forall119906 isin 119882

1119901(Ω) (19)

with 119888 = 119888 (Ω 119899 119901 119903)If 119892 isin 119872

119905(Ω) with 119905 gt 119901 if 119901 = 1198992 then the operator in

(17) is bounded from1198822119901(Ω) to 119871119901(Ω) Moreover there exists

a constant 1198881015840 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 119871119901(Ω) le 119888

101584010038171003817100381710038171198921003817100381710038171003817119872119905(Ω)1199061198822119901(Ω) forall119906 isin 119882

2119901(Ω) (20)

with 1198881015840 = 1198881015840(Ω 119899 119901 119905)

3 The Variational Problem

Consider in an unbounded open subset Ω of R119899 119899 ge 2 thesecond-order linear differential operator in divergence form

119871 = minus

119899

sum

119894119895=1

120597

120597119909119895(119886119894119895

120597

120597119909119894+ 119889119895) +

119899

sum

119894=1

119887119894120597

120597119909119894+ 119888 (21)

Assume that the leading coefficients satisfy the hypotheses

119886119894119895 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(ℎ1)

For the lower order terms coefficients suppose that

119887119894 119889119894 isin 1198722119905120582

119900(Ω) 119894 = 1 119899

119888 isin 119872119905120582(Ω)

with 119905 gt 1 and 120582 = 0 if 119899 = 2

with 119905 isin ]1119899

2] and 120582 = 119899 minus 2119905 if 119899 gt 2

(ℎ2)

Furthermore let one of the following sign assumptions holdtrue

119888 minus

119899

sum

119894=1

(119889119894)119909119894ge 120583 (ℎ3)

or

119888 minus

119899

sum

119894=1

(119887119894)119909119894ge 120583 (ℎ4)

in the distributional sense onΩ with 120583 positive constant

We are interested in the study of the Dirichlet problem

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 119882minus12

(Ω)

(22)

(ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2) and (ℎ4) being satisfiedIt is natural to associate to 119871 the bilinear form

119886 (119906 V) = intΩ

(

119899

sum

119894119895=1

(119886119894119895119906119909119894+ 119889119895119906) V119909119895

+(

119899

sum

119894=1

119887119894119906119909119894+ 119888119906) V)119889119909

(23)

119906 V isin∘

11988212(Ω) and observe that in view of Theorem 1 the

form 119886 is continuous on∘

11988212(Ω) times

∘

11988212(Ω) and so the

operator 119871 ∘

11988212(Ω) rarr 119882

minus12(Ω) is continuous too

Let us start collecting some preliminary results concern-ing the existence and uniqueness of the solution of problem(22) as well as some a priori estimates For the case whereassumptions (ℎ1)ndash(ℎ3) are taken into account and for 119899 = 2we refer to [2] while for 119899 ge 3 details can be found in [3]If (ℎ1) (ℎ2) and (ℎ4) hold true the results are proved in themore recent [5]

Theorem 2 Under hypotheses (ℎ1)ndash(ℎ3) (or (ℎ1) (ℎ2) and(ℎ4)) problem (22) is uniquely solvable and its solution 119906

satisfies the estimate

11990611988212(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817119882minus12(Ω) (24)

where 119862 is a constant depending on 119899 119905 ] 120583 ||119887119894 minus 119889119894||1198722119905120582(Ω)119894 = 1 119899

The next step in our analysis is to achieve an 119871119901-estimate119901 gt 2 for the solution of (22) (see Theorem 8) This requiressome additional hypotheses on the regularity of the set andon the datum 119891 and some preparatory results that essentiallyrely on the introduction of certain auxiliary functions119906119904 usedfor the first time by Bottaro and Marina in [1] and employedin the framework of Morrey type spaces in [3] Let us givetheir definition and recall some useful properties

Let ℎ isin R+ cup +infin and 119896 isin R with 0 le 119896 le ℎ For each119905 isin R we set

119866119896ℎ (119905) =

119905 minus 119896 if 119905 gt 119896

0 if minus 119896 le 119905 le 119896 if ℎ = +infin

119905 + 119896 if 119905 lt minus119896

119866119896ℎ (119905) = 119866119896infin (119905) minus 119866ℎinfin (119905) if ℎ isin R+

(25)

Lemma 3 Let 119892 isin 119872119902120582

119900(Ω) 119906 isin

∘

11988212(Ω) and 120576 isin R+ Then

there exist 119903 isin N and 1198961 119896119903 isin R with 0 = 119896119903 lt 119896119903minus1 lt

sdot sdot sdot lt 1198961 lt 1198960 = +infin such that set

119906119904 = 119866119896119904119896119904minus1(119906) 119904 = 1 119903 (26)

4 Abstract and Applied Analysis

one has 1199061 119906119903 isin∘

11988212(Ω) and

10038171003817100381710038171003817119892120594supp (119906119904)119909

10038171003817100381710038171003817119872119902120582(Ω)le 120576 119904 = 1 119903 (27)

10038161003816100381610038161199061199041003816100381610038161003816 le |119906| 119904 = 1 119903 (28)

1199061 + sdot sdot sdot + 119906119903 = 119906 (29)

119903 le 119888 (30)

with 119888 = 119888 (120576 119902 119892119872119902120582(Ω)

) positive constant

In order to prove a fundamental preliminary estimateobtained for 119901 gt 2 (seeTheorem 7) we need to take productsinvolving the above defined functions 119906119904 as test functions inthe variational formulation of our problem (23) To be moreprecise in the first set of hypotheses ((ℎ1)ndash(ℎ3)) the testfunctions needed are |119906|119901minus2119906119904 The following result ensuresthat these functions effectively belong to

∘

11988212(Ω)

Lemma 4 If Ω has the uniform 1198621-regularity property then

for every 119906 isin∘

11988212(Ω) cap 119871

infin(Ω) and for any 119901 isin ]2 +infin[ one

has

|119906|119901minus2

119906119904 isin∘

11988212(Ω) 119904 = 1 119903 (31)

Lemma 4 whose rather technical proof can be found in [4]is a generalization of a known result by Stampacchia (see[18] or [19] for details) obtained within the framework ofthe generalization of the study of certain elliptic equationsin divergence form with discontinuous coefficients on abounded open subset of R119899 to some problems arisingfor harmonic or subharmonic functions in the theory ofpotential

Once achieved (31) always in [4] we could prove thenext lemma Let 119906119904 be the functions of Lemma 3 obtainedin correspondence of a given 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of

119892 = sum119899

119894=1|119887119894 minus 119889119894| and of a positive real number 120576 specified

in the proof of Lemma 41 of [4] One has the following

Lemma 5 Let 119886 be the bilinear form defined in (23) If Ωhas the uniform119862

1-regularity property under hypotheses (ℎ1)ndash(ℎ3) there exists a constant 119862 isin R+ such that

intΩ

|119906|119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(32)

where 119862 depends on 119904 ] 120583

If we consider the second set of hypotheses ((ℎ1) (ℎ2)and (ℎ4)) the test functions required in (23) are the prod-ucts |119906119904|

119901minus2119906119904 obtained in correspondence of a fixed 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of 119892 = sum

119899

119894=1|119889119894 minus 119887119894| and of a positive real

number 120576 specified in the proof of Lemma 41 of [5] In thislast case and if Ω has the uniform 119862

1-regularity property aresult of [20] applies giving that |119906119904|

119901minus2119906119904 isin

∘

11988212(Ω) for any

119901 gt 2 119904 = 1 119903 Hence in [5] we could show the result

Lemma 6 Let 119886 be the bilinear form in (23) If Ω has theuniform 119862

1-regularity property under hypotheses (ℎ1) (ℎ2)and (ℎ4) there exists a constant 119862 isin R+ such that

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(33)

where 119862 depends on 119904 119903 ] 120583

The two lemmas just stated put us in a position to provethe following preliminary 119871

119901-a priori estimate 119901 gt 2 inboth sets of hypotheses see also [4 5] We stress that here werequire that both the datum119891 and the solution119906 are bounded

Theorem 7 Under hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2) and(ℎ4) and if Ω has the uniform 119862

1-regularity property 119891 is in1198712(Ω) cap 119871

infin(Ω) and the solution 119906 of (22) is in

∘

11988212(Ω) cap

119871infin(Ω) then 119906 isin 119871119901(Ω) and

119906 119871119901(Ω) le C10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) forall119901 isin ]2 +infin[ (34)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof Fix 119901 isin ]2 +infin[ We provide two different proofs inthe cases that hypotheses (ℎ3) or (ℎ4) hold true

Let (ℎ1)ndash(ℎ3) be satisfied We consider the functions 119906119904119904 = 1 119903 obtained in correspondence of the solution 119906 andof 119892 = sum

119899

119894=1|1198891 minus 119887119894| and of 120576 as in Lemma 41 of [4] In view

of (29) we get

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880intΩ

|119906|119901minus2119903

sum

119904=1

((119906119904)2

119909+ 1199062

119904) 119889119909

(35)

with 1198880 = 1198880(119903)Hence (32) entails that

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880

119903

sum

119904=1

119862119904

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

le 119862

119903

sum

119904=1

119886 (119906 |119906|119901minus2

119906119904)

(36)

with 119862119904 = 119862119904(119904 ] 120583) and 119862 = 119862(119903 ] 120583)From the linearity of 119886 (29) and (30) we have then

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909 le 119862119886 (119906 |119906|

119901minus2119906) (37)

with 119862 = 119862(119899 119905 119901 ] 120583 ||119887119894 minus 119889119894||1198722119905120582(Ω))

Abstract and Applied Analysis 5

Using this last inequality and Holder inequality weconclude our proof since

119906119901

119871119901(Ω)le intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 119862119886 (119906 |119906|119901minus2

119906) = 119862intΩ

119891|119906|119901minus2

119906119889119909

le 119862intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909 le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(38)

If (ℎ1) (ℎ2) and (ℎ4) hold we consider again thefunctions 119906119904 119904 = 1 119903 obtained in correspondence ofthe solution 119906 and of 119892 as in the previous case and of 120576 asin Lemma 41 of [5] In this second case easy computationstogether with (29) give

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901119889119909 (39)

with 1198880 = 1198880(119903 119901)Thus from (33) we deduce that

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

119862s

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

le 1198881

119903

sum

119904=1

119886 (11990610038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904)

(40)

with 119862119904 = 119862119904(119904 119903 ] 120583) and 1198881 = 1198881(119903 119901 ] 120583)Hence by (28) and Holder inequality we obtain

119906119901

119871119901(Ω)le 1198881

119903

sum

119904=1

intΩ

11989110038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904119889119909

le 1199031198881intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909

le 119903119888110038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(41)

This ends the proof in view of (30)

In the later paper [6] estimate (34) has been improveddropping the hypotheses on the boundedness of 119891 and 119906 bymeans of the theorem below

Theorem 8 Assume that hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]2 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (42)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

The proof which is different according to hypothesis (ℎ3)or (ℎ4) is essentially performed into two steps In the firststep we show some regularity results exploiting a technique

introduced by Miranda in [21] Namely we prove that if 119906 isin∘

11988212(Ω) is the solution of (22) with119891 isin 119871

2(Ω)cap119871

infin(Ω) then

the datum 119891 being more regular one also has 119906 isin 119871infin(Ω)

Thus Theorem 7 applies giving that 119906 isin 119871119901(Ω) and satisfies

(34) The second step consists in considering a datum 119891 isin

1198712(Ω) cap 119871

119901(Ω) and then one can conclude by means of some

approximation arguments see also [16]Finally in [6] we prove the main result that is the

claimed 119871119901-bound 119901 gt 1 To this aim a further assumptionon the leading coefficients is required

119886119894119895 = 119886119895119894 119894 119895 = 1 119899 (ℎ0)

Then one has the following

Theorem 9 Assume that hypotheses (ℎ0)ndash(ℎ3) or (ℎ0) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]1 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (43)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof For 119901 ge 2 Theorems 2 and 8 already prove the resultIt remains to show it for 1 lt 119901 lt 2

We assume that hypotheses (ℎ0)ndash(ℎ3) hold true Underhypotheses (ℎ0) (ℎ2) and (ℎ4) a similar argument withsuitable modifications can be used (we refer the reader to [6]for the details)

Let us define the bilinear form

119886lowast(119908 V) = 119886 (V 119908) 119908 V isin

∘

11988212(Ω) (44)

By (ℎ0) one has

119886lowast(119908 V)

= intΩ

(

119899

sum

119894119895=1

(119886119894119895119908119909119894+119887119895119908) V119909119895+(

119899

sum

119894=1

119889119894119908119909119894+119888119908) V)119889119909

(45)

Now consider the problem

119908 isin∘

11988212(Ω)

119886lowast(119908 V) = int

Ω

119892V 119889119909 119892 isin 1198712(Ω) cap 119871

1199011015840

(Ω)

(46)

where since 1 lt 119901 lt 2 one gets 1199011015840 = 119901(119901 minus 1) gt 2As a consequence of Theorem 2 (in the second set of

hypotheses) the solution 119908 of (46) exists and is uniqueFurthermore byTheorem 8 (in the second set of hypotheses)one also has

1199081198711199011015840

(Ω)le 119862

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω) (47)

Hence if we denote by 119906 the solution of

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 1198712(Ω) cap 119871

119901(Ω)

(48)

6 Abstract and Applied Analysis

which exists and is unique in view of Theorem 2 (in the firstset of hypotheses) we obtain

intΩ

119892119906 119889119909 = 119886lowast(119908 119906) = 119886 (119906 119908) = int

Ω

119891119908 119889119909

le10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119908 119871119901

1015840

(Ω)le 119862

10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω)

(49)

Finally taking 119892 = |119906|119901minus1 sign 119906 in (49) we get the claimed

result

4 Non-Variational Problems

In this section we show two applications of ourmain estimate(43)

To this aim let 119901 gt 1 and assume that

Ω has the uniform 11986211-regularity property (ℎ

1015840

0)

Consider then the non-variational differential operator

119871 = minus

119899

sum

119894119895=1

1198861198941198951205972

120597119909119894120597119909119895+

119899

sum

119894=1

119886119894120597

120597119909119894+ 119886 (50)

with the following conditions on the leading coefficients

119886119894119895 = 119886119895119894 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(119886119894119895)119909ℎisin 119872119902120582

119900(Ω) 119894 119895 ℎ = 1 119899 with

119902 gt 2 120582 = 0 for 119899 = 2

119902 isin ] 2 119899 [ 120582 = 119899 minus 119902 for 119899 gt 2

(ℎ1015840

1)

Suppose that the lower order terms are such that

119886119894 isin 119872119903

119900(Ω) 119894 = 1 119899 with

119903 gt 2 if 119901 le 2 119903 = 119901 if 119901 gt 2 for 119899 = 2

119903 ge 119901 119903 ge 119899 with 119903 gt 119901 if 119901 = 119899 for 119899 gt 2

(ℎ1015840

2)

119886 isin 119905(Ω) with

119905 = 119901 for 119899 = 2

119905 ge 119901 119905 ge119899

2 with 119905 gt 119901 if 119901 =

119899

2for 119899 gt 2

ess infΩ

119886 = 1198860 gt 0

(ℎ1015840

3)

In view of Theorem 1 under the assumptions (ℎ10158400)ndash(ℎ10158403) the

operator 119871 1198822119901(Ω) rarr 119871119901(Ω) is bounded

The first application is contained in Theorem 32 andCorollary 33 of [7] (see also [22] where the case 119901 = 2 isconsidered) and reads as follows

Theorem 10 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901(Ω) le 1198881003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901(Ω)

forall119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω) (51)

with 119888= 119888(Ω 119899 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) 120590119902120582

119900[(119886119894119895)119909ℎ

] 120590119903

119900[119886119894]

119905[119886]

1198860)Moreover the problem

119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω)

119871119906 = 119891 119891 isin 119871119901(Ω)

(52)

is uniquely solvable

The nodal point in achieving these results consists in theexistence of the derivatives of the 119886119894119895 Indeed this consentsto rewrite the operator 119871 in divergence form and exploit(43) in order to obtain an estimate as that in (51) but formore regular functions Then one can prove (51) by meansof an approximation argument Estimate (51) immediatelytakes to the solvability of problem (52) via a straightforwardapplication of the method of continuity along a parametersee for instance [23] and by the already known solvabilityof an opportune auxiliary problem

As second application of (43) we obtain in [8] ananalogous of Theorem 10 in a weighted framework Namelywe consider a weight function 120588119904 that is a power of a function120588 of class 1198622(Ω) such that 120588 Ω rarr R+ and

sup119909isinΩ

1003816100381610038161003816120597120572120588 (119909)

1003816100381610038161003816

120588 (119909)lt +infin forall |120572| le 2

lim|119909|rarr+infin

(120588 (119909) +1

120588 (119909)) = +infin

lim|119909|rarr+infin

120588119909(119909) + 120588

119909119909(119909)

120588 (119909)= 0

(53)

For instance one can think of 120588 as the function

120588 (119909) = (1 + |119909|2)119905

119905 isin R 0 (54)

For 119896 isin N0 119901 isin [1 +infin[ and 119904 isin R and given 120588 satisfying(53) we define the weighted Sobolev space 119882119896119901

119904(Ω) as the

space of distributions 119906 onΩ such that

119906119882119896119901

119904 (Ω)= sum

|120572|le119896

10038171003817100381710038171205881199041205971205721199061003817100381710038171003817 119871119901(Ω) lt +infin (55)

endowed with the norm in (55) Furthermore we denote theclosure of 119862infin

∘(Ω) in119882119896119901

119904(Ω) by

∘

119882119896119901

119904(Ω) and put1198820119901

119904(Ω) =

119871119901

119904(Ω)In Theorems 42 and 52 of [8] we showed the following

Theorem 11 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901

119904 (Ω)le 119888

1003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901119904 (Ω)

forall119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

(56)

Abstract and Applied Analysis 7

with 119888 = 119888(Ω 119899 119904 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) ||119886119894||119872119903(Ω) 120590119902120582119900 [(119886119894119895)119909ℎ]120590119903

119900[119886119894]

119905[119886] 1198860)

Moreover the problem

119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

119871119906 = 119891 119891 isin 119871119901

119904(Ω)

(57)

is uniquely solvable

One of the main tools in the proof of Theorem 11 is givenby the existence of a topological isomorphism from119882

119896119901

119904(Ω)

to119882119896119901(Ω) and from∘

119882119896119901

119904(Ω) to

∘

119882119896119901(Ω)This isomorphism

consents to deduce by the non-weighted bound in (51) thecorresponding weighted estimate in (56) taking into accountalso the imbedding results of Theorem 1 The existence anduniqueness of the solution of problem (57) follow then as inthe previous case from a direct application of the method ofcontinuity along a parameter by the solvability of a suitableauxiliary problem

References

[1] G Bottaro and M E Marina ldquoProblema di Dirichlet perequazioni ellittiche di tipo variazionale su insiemi non limitatirdquoBollettino dellrsquoUnione Matematica Italiana Serie 4 vol 8 pp46ndash56 1973

[2] M Transirico and M Troisi ldquoEquazioni ellittiche del secondoordine a coefficienti discontinui e di tipo variazionale in apertinon limitatirdquo Bollettino dellrsquoUnioneMatematica Italiana Serie 7vol 2 no 1 pp 385ndash398 1988

[3] M Transirico M Troisi and A Vitolo ldquoSpaces of Morreytype and elliptic equations in divergence form on unboundeddomainsrdquo Bollettino dellrsquoUnione Matematica Italiana Serie 7vol 9 no 1 pp 153ndash174 1995

[4] S Monsurro and M Transirico ldquoAn 119871119901-estimate for weak

solutions of elliptic equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 376179 15 pages 2012

[5] S Monsurro and M Transirico ldquoDirichlet problem for diver-gence form elliptic equations with discontinuous coefficientsrdquoBoundary Value Problems vol 2012 20 pages 2012

[6] S Monsurro and M Transirico ldquoA priori bounds in 119871119901 for

solutions of elliptic equations in divergence formrdquo Bulletin desSciences Mathematiques In press

[7] S Monsurro and M Transirico ldquoA 1198822119901-estimate for a class

of elliptic operatorsrdquo International Journal of Pure and AppliedMathematics vol 83 no 4 pp 489ndash499 2013

[8] S Monsurro and M Transirico ldquoA weighted 1198822119901-bound

for a class of elliptic operatorsrdquo Journal of Inequalities andApplications vol 2013 article 263 2013

[9] C Miranda ldquoSulle equazioni ellittiche del secondo ordine ditipo non variazionale a coefficienti discontinuirdquo Annali diMatematica Pura ed Applicata vol 63 no 1 pp 353ndash386 1963

[10] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines nonbornesrdquo Atti della Accademia Nazionale dei Lincei RendicontiClasse di Scienze Fisiche Matematiche e Naturali Serie VIII vol78 no 5 pp 205ndash212 1985

[11] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines

non bornes IIrdquo Atti della Accademia Nazionale dei LinceiRendiconti Classe di Scienze Fisiche Matematiche e NaturaliSerie VIII vol 79 no 6 pp 178ndash183 1985

[12] P Donato and D Giachetti ldquoQuasilinear elliptic equations withquadratic growth in unbounded domainsrdquo Nonlinear AnalysisTheoryMethods ampApplications vol 10 no 8 pp 791ndash804 1986

[13] L Caso P Cavaliere and M Transirico ldquoAn existence resultfor elliptic equations with VMO-coefficientsrdquo Journal of Mathe-matical Analysis and Applications vol 325 no 2 pp 1095ndash11022007

[14] S Boccia S Monsurro and M Transirico ldquoElliptic equationsin weighted Sobolev spaces on unbounded domainsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2008 Article ID 582435 12 pages 2008

[15] S Boccia S Monsurro and M Transirico ldquoSolvability of theDirichlet problem for elliptic equations in weighted Sobolevspaces on unbounded domainsrdquo Boundary Value Problems vol2008 Article ID 901503 13 pages 2008

[16] L Caso R DrsquoAmbrosio and S Monsurro ldquoSome remarks onspaces of Morrey typerdquo Abstract and Applied Analysis vol 2010Article ID 242079 22 pages 2010

[17] P Cavaliere M Longobardi and A Vitolo ldquoImbedding esti-mates and elliptic equations with discontinuous coefficients inunbounded domainsrdquo LeMatematiche vol 51 no 1 pp 87ndash1041996

[18] G Stampacchia ldquoLe probleme de Dirichlet pour les equationselliptiques du second ordre a coefficients discontinusrdquo Annalesde lrsquoInstitut Fourier vol 15 pp 189ndash258 1965

[19] G Stampacchia Equations Elliptiques du Second Ordre a Coef-ficients Discontinus Seminaire de Mathematiques Superieuresno 16 (Ete 1965) Les Presses de lrsquoUniversite de MontrealMontreal Canada 1966

[20] L Caso P Cavaliere and M Transirico ldquoSolvability of theDirichlet problem in1198822119901 for elliptic equations with discontin-uous coefficients in unbounded domainsrdquo Le Matematiche vol57 no 2 pp 287ndash302 2002

[21] C Miranda ldquoAlcune osservazioni sulla maggiorazione in 119871]119901

delle soluzioni deboli delle equazioni ellittiche del secondoordinerdquoAnnali diMatematica Pura ed Applicata vol 61 pp 151ndash169 1963

[22] S Monsurro M Salvato and M Transirico ldquo11988222 a prioribounds for a class of elliptic operatorsrdquo International Journalof Differential Equations vol 2011 Article ID 572824 17 pages2011

[23] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 Springer Berlin Germany2nd edition 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

2 Abstract and Applied Analysis

for any bounded solution 119906 of (2) and for every 119901 gt 2This can be done taking into account two different signhypotheses namely (5) and the less common

119888 minus

119899

sum

119894=1

(119887119894)119909119894ge 120583 120583 isin R+ (8)

Successively in [6] we deepen the study begun in [4 5]showing that to a bounded datum 119891 isin 119871

2(Ω) it corresponds

a bounded solution 119906 This allows us to prove by means of anapproximation argument that if 119891 belongs to 1198712(Ω) cap 119871119901(Ω)119901 gt 2 then the solution is in 119871

119901(Ω) too and verifies (7)

Putting together the two preliminary 119871119901-estimates 119901 gt 2obtained under the different sign assumptions and adding thefurther hypothesis that the 119886119894119895 are also symmetric by meansof a duality argument we finally obtain (7) for 119901 gt 1 for eachsign hypothesis assuming no boundedness of the solutionand for 119891 isin 119871

2(Ω) cap 119871

119901(Ω)

To conclude we provide two applications of our final119871119901-bound 119901 gt 1 recalling the results of [7 8] where our

estimate plays a fundamental role in the study of certainweighted and non-weighted non-variational problems withleading coefficients satisfying hypotheses of Mirandarsquos type(see [9])Thenodal point in this analysis is the existence of thederivatives of the leading coefficients that allows us to rewritethe involved operator in variational form and avail ourselvesof the above-mentioned a priori bound

Always in the framework of unbounded domains thestudy of different variational problems can be found in[10 11] Quasilinear elliptic equations with quadratic growthhave been considered in [12] In [13ndash15] a very generalweighted case with principal coefficients having vanishingmean oscillation has been taken into account

2 A Class of Spaces of Morrey Type

In this section we recall the definitions and the main prop-erties of a certain class of spaces of Morrey type where thecoefficients of our operators belong These spaces generalizethe classical notion of Morrey spaces to unbounded domainsand were introduced for the first time in [3] see also [16]for some details Thus from now on let Ω be an unboundedopen subset of R119899 119899 ge 2 By Σ(Ω) we denote the 120590-algebraof all Lebesgue measurable subsets of Ω For 119864 isin Σ(Ω) 120594

119864

is its characteristic function |119864| its Lebesgue measure and119864(119909 119903) = 119864 cap 119861(119909 119903) (119909 isin R119899 119903 isin R+) where 119861(119909 119903) is theopen ball with center in 119909 and radius 119903 The class ofrestrictions to Ω of functions 120577 isin 119862

infin

∘(R119899) is D(Ω) For

119902 isin [1 +infin[ 119871119902loc (Ω) is the class of all functions 119892 Ω rarr R

such that 120577 119892 isin 119871119902(Ω) for any 120577 isin D(Ω)For 119902 isin [1 +infin[ and 120582 isin [0 119899[ the space of Morrey type

119872119902120582(Ω) is made up of all the functions 119892 in 119871119902loc (Ω) such that

10038171003817100381710038171198921003817100381710038171003817119872119902120582(Ω) = sup

120591isin]01]

119909isinΩ

120591minus1205821199021003817100381710038171003817119892

1003817100381710038171003817 119871119902(Ω(119909120591)) lt +infin (9)

equipped with the norm defined in (9)

The closures of 119862infin∘(Ω) and 119871infin(Ω) in119872119902120582(Ω) are deno-

ted by119872119902120582∘ (Ω) and 119902120582(Ω) respectivelyThe following inclusion holds true

119872119902120582

∘(Ω) sub

119902120582(Ω) (10)

Moreover

119872119902120582

(Ω) sube 11987211990201205820 (Ω) if 1199020 le 119902

1205820 minus 119899

1199020le120582 minus 119899

119902

(11)

We put 119872119902(Ω) = 1198721199020(Ω) 119902(Ω) =

1199020(Ω) and

119872119902

∘ (Ω) = 1198721199020

∘ (Ω)Now let us define the moduli of continuity of functions

belonging to 119902120582(Ω) or 119872119902120582∘ (Ω) For ℎ isin R+ and 119892 isin

119872119902120582(Ω) we set

119865 [119892] (ℎ) = sup119864isinΣ(Ω)

sup119909isinΩ|119864(1199091)|le1ℎ

10038171003817100381710038171198921205941198641003817100381710038171003817119872119902120582(Ω) (12)

Given a function 119892 isin 119872119902120582(Ω) the following characteriza-

tions hold

119892 isin 119902120582

(Ω) lArrrArr limℎrarr+infin

119865 [119892] (ℎ) = 0

119892 isin 119872119902120582

∘(Ω)

lArrrArr limℎrarr+infin

(119865 [119892] (ℎ) +1003817100381710038171003817 (1 minus 120577ℎ) 119892

1003817100381710038171003817119872119902120582(Ω)) = 0

(13)

where 120577ℎ denotes a function of class 119862infin119900(119877119899) such that

0 le 120577ℎ le 1

120577ℎ|119861(0ℎ)

= 1

supp 120577ℎ sub 119861 (0 2ℎ)

(14)

Thus if 119892 is a function in 119902120582(Ω) amodulus of continuity of119892 in 119902120582(Ω) is a map 119902120582[119892] R+ rarr R+ such that

119865 [119892] (ℎ) le 119902120582

[119892] (ℎ)

limℎrarr+infin

119902120582

[119892] (ℎ) = 0(15)

While if 119892 belongs to119872119902120582119900(Ω) amodulus of continuity of 119892 in

119872119902120582

119900(Ω) is an application 120590119902120582

119900[119892] R+ rarr R+ such that

119865 [119892] (ℎ) +1003817100381710038171003817 (1 minus 120577ℎ) 119892

1003817100381710038171003817119872119902120582(Ω) le 120590119902120582

119900[119892] (ℎ)

limℎrarr+infin

120590119902120582

119900[119892] (ℎ) = 0

(16)

We finally recall two results of [4 7] obtained adapting toour needs a more general theorem proved in [17] providingthe boundedness and some embedding estimates for themultiplication operator

119906 997888rarr 119892119906 (17)

where the function 119892 belongs to suitable spaces of Morreytype

Abstract and Applied Analysis 3

Theorem 1 If 119892 isin 119872119902120582(Ω) with 119902 gt 2 and 120582 = 0 if 119899 = 2

and 119902 isin ]2 119899] and 120582 = 119899 minus 119902 if 119899 gt 2 then the operator in(17) is bounded from

∘

11988212(Ω) to 1198712(Ω) Moreover there exists

a constant 119862 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 1198712(Ω) le 119862

10038171003817100381710038171198921003817100381710038171003817119872119902120582(Ω)11990611988212(Ω) forall119906 isin

∘

11988212(Ω) (18)

with 119862 = 119862(119899 119902)Let 119901 gt 1 and 119903 119905 isin [119901 +infin[ If Ω is an open subset of R119899

having the cone property and 119892 isin 119872119903(Ω) with 119903 gt 119901 if 119901 = 119899

then the operator in (17) is bounded from 1198821119901(Ω) to 119871119901(Ω)

Moreover there exists a constant 119888 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 119871119901(Ω) le 119888

10038171003817100381710038171198921003817100381710038171003817119872119903(Ω)1199061198821119901(Ω) forall119906 isin 119882

1119901(Ω) (19)

with 119888 = 119888 (Ω 119899 119901 119903)If 119892 isin 119872

119905(Ω) with 119905 gt 119901 if 119901 = 1198992 then the operator in

(17) is bounded from1198822119901(Ω) to 119871119901(Ω) Moreover there exists

a constant 1198881015840 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 119871119901(Ω) le 119888

101584010038171003817100381710038171198921003817100381710038171003817119872119905(Ω)1199061198822119901(Ω) forall119906 isin 119882

2119901(Ω) (20)

with 1198881015840 = 1198881015840(Ω 119899 119901 119905)

3 The Variational Problem

Consider in an unbounded open subset Ω of R119899 119899 ge 2 thesecond-order linear differential operator in divergence form

119871 = minus

119899

sum

119894119895=1

120597

120597119909119895(119886119894119895

120597

120597119909119894+ 119889119895) +

119899

sum

119894=1

119887119894120597

120597119909119894+ 119888 (21)

Assume that the leading coefficients satisfy the hypotheses

119886119894119895 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(ℎ1)

For the lower order terms coefficients suppose that

119887119894 119889119894 isin 1198722119905120582

119900(Ω) 119894 = 1 119899

119888 isin 119872119905120582(Ω)

with 119905 gt 1 and 120582 = 0 if 119899 = 2

with 119905 isin ]1119899

2] and 120582 = 119899 minus 2119905 if 119899 gt 2

(ℎ2)

Furthermore let one of the following sign assumptions holdtrue

119888 minus

119899

sum

119894=1

(119889119894)119909119894ge 120583 (ℎ3)

or

119888 minus

119899

sum

119894=1

(119887119894)119909119894ge 120583 (ℎ4)

in the distributional sense onΩ with 120583 positive constant

We are interested in the study of the Dirichlet problem

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 119882minus12

(Ω)

(22)

(ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2) and (ℎ4) being satisfiedIt is natural to associate to 119871 the bilinear form

119886 (119906 V) = intΩ

(

119899

sum

119894119895=1

(119886119894119895119906119909119894+ 119889119895119906) V119909119895

+(

119899

sum

119894=1

119887119894119906119909119894+ 119888119906) V)119889119909

(23)

119906 V isin∘

11988212(Ω) and observe that in view of Theorem 1 the

form 119886 is continuous on∘

11988212(Ω) times

∘

11988212(Ω) and so the

operator 119871 ∘

11988212(Ω) rarr 119882

minus12(Ω) is continuous too

Let us start collecting some preliminary results concern-ing the existence and uniqueness of the solution of problem(22) as well as some a priori estimates For the case whereassumptions (ℎ1)ndash(ℎ3) are taken into account and for 119899 = 2we refer to [2] while for 119899 ge 3 details can be found in [3]If (ℎ1) (ℎ2) and (ℎ4) hold true the results are proved in themore recent [5]

Theorem 2 Under hypotheses (ℎ1)ndash(ℎ3) (or (ℎ1) (ℎ2) and(ℎ4)) problem (22) is uniquely solvable and its solution 119906

satisfies the estimate

11990611988212(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817119882minus12(Ω) (24)

where 119862 is a constant depending on 119899 119905 ] 120583 ||119887119894 minus 119889119894||1198722119905120582(Ω)119894 = 1 119899

The next step in our analysis is to achieve an 119871119901-estimate119901 gt 2 for the solution of (22) (see Theorem 8) This requiressome additional hypotheses on the regularity of the set andon the datum 119891 and some preparatory results that essentiallyrely on the introduction of certain auxiliary functions119906119904 usedfor the first time by Bottaro and Marina in [1] and employedin the framework of Morrey type spaces in [3] Let us givetheir definition and recall some useful properties

Let ℎ isin R+ cup +infin and 119896 isin R with 0 le 119896 le ℎ For each119905 isin R we set

119866119896ℎ (119905) =

119905 minus 119896 if 119905 gt 119896

0 if minus 119896 le 119905 le 119896 if ℎ = +infin

119905 + 119896 if 119905 lt minus119896

119866119896ℎ (119905) = 119866119896infin (119905) minus 119866ℎinfin (119905) if ℎ isin R+

(25)

Lemma 3 Let 119892 isin 119872119902120582

119900(Ω) 119906 isin

∘

11988212(Ω) and 120576 isin R+ Then

there exist 119903 isin N and 1198961 119896119903 isin R with 0 = 119896119903 lt 119896119903minus1 lt

sdot sdot sdot lt 1198961 lt 1198960 = +infin such that set

119906119904 = 119866119896119904119896119904minus1(119906) 119904 = 1 119903 (26)

4 Abstract and Applied Analysis

one has 1199061 119906119903 isin∘

11988212(Ω) and

10038171003817100381710038171003817119892120594supp (119906119904)119909

10038171003817100381710038171003817119872119902120582(Ω)le 120576 119904 = 1 119903 (27)

10038161003816100381610038161199061199041003816100381610038161003816 le |119906| 119904 = 1 119903 (28)

1199061 + sdot sdot sdot + 119906119903 = 119906 (29)

119903 le 119888 (30)

with 119888 = 119888 (120576 119902 119892119872119902120582(Ω)

) positive constant

In order to prove a fundamental preliminary estimateobtained for 119901 gt 2 (seeTheorem 7) we need to take productsinvolving the above defined functions 119906119904 as test functions inthe variational formulation of our problem (23) To be moreprecise in the first set of hypotheses ((ℎ1)ndash(ℎ3)) the testfunctions needed are |119906|119901minus2119906119904 The following result ensuresthat these functions effectively belong to

∘

11988212(Ω)

Lemma 4 If Ω has the uniform 1198621-regularity property then

for every 119906 isin∘

11988212(Ω) cap 119871

infin(Ω) and for any 119901 isin ]2 +infin[ one

has

|119906|119901minus2

119906119904 isin∘

11988212(Ω) 119904 = 1 119903 (31)

Lemma 4 whose rather technical proof can be found in [4]is a generalization of a known result by Stampacchia (see[18] or [19] for details) obtained within the framework ofthe generalization of the study of certain elliptic equationsin divergence form with discontinuous coefficients on abounded open subset of R119899 to some problems arisingfor harmonic or subharmonic functions in the theory ofpotential

Once achieved (31) always in [4] we could prove thenext lemma Let 119906119904 be the functions of Lemma 3 obtainedin correspondence of a given 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of

119892 = sum119899

119894=1|119887119894 minus 119889119894| and of a positive real number 120576 specified

in the proof of Lemma 41 of [4] One has the following

Lemma 5 Let 119886 be the bilinear form defined in (23) If Ωhas the uniform119862

1-regularity property under hypotheses (ℎ1)ndash(ℎ3) there exists a constant 119862 isin R+ such that

intΩ

|119906|119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(32)

where 119862 depends on 119904 ] 120583

If we consider the second set of hypotheses ((ℎ1) (ℎ2)and (ℎ4)) the test functions required in (23) are the prod-ucts |119906119904|

119901minus2119906119904 obtained in correspondence of a fixed 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of 119892 = sum

119899

119894=1|119889119894 minus 119887119894| and of a positive real

number 120576 specified in the proof of Lemma 41 of [5] In thislast case and if Ω has the uniform 119862

1-regularity property aresult of [20] applies giving that |119906119904|

119901minus2119906119904 isin

∘

11988212(Ω) for any

119901 gt 2 119904 = 1 119903 Hence in [5] we could show the result

Lemma 6 Let 119886 be the bilinear form in (23) If Ω has theuniform 119862

1-regularity property under hypotheses (ℎ1) (ℎ2)and (ℎ4) there exists a constant 119862 isin R+ such that

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(33)

where 119862 depends on 119904 119903 ] 120583

The two lemmas just stated put us in a position to provethe following preliminary 119871

119901-a priori estimate 119901 gt 2 inboth sets of hypotheses see also [4 5] We stress that here werequire that both the datum119891 and the solution119906 are bounded

Theorem 7 Under hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2) and(ℎ4) and if Ω has the uniform 119862

1-regularity property 119891 is in1198712(Ω) cap 119871

infin(Ω) and the solution 119906 of (22) is in

∘

11988212(Ω) cap

119871infin(Ω) then 119906 isin 119871119901(Ω) and

119906 119871119901(Ω) le C10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) forall119901 isin ]2 +infin[ (34)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof Fix 119901 isin ]2 +infin[ We provide two different proofs inthe cases that hypotheses (ℎ3) or (ℎ4) hold true

Let (ℎ1)ndash(ℎ3) be satisfied We consider the functions 119906119904119904 = 1 119903 obtained in correspondence of the solution 119906 andof 119892 = sum

119899

119894=1|1198891 minus 119887119894| and of 120576 as in Lemma 41 of [4] In view

of (29) we get

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880intΩ

|119906|119901minus2119903

sum

119904=1

((119906119904)2

119909+ 1199062

119904) 119889119909

(35)

with 1198880 = 1198880(119903)Hence (32) entails that

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880

119903

sum

119904=1

119862119904

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

le 119862

119903

sum

119904=1

119886 (119906 |119906|119901minus2

119906119904)

(36)

with 119862119904 = 119862119904(119904 ] 120583) and 119862 = 119862(119903 ] 120583)From the linearity of 119886 (29) and (30) we have then

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909 le 119862119886 (119906 |119906|

119901minus2119906) (37)

with 119862 = 119862(119899 119905 119901 ] 120583 ||119887119894 minus 119889119894||1198722119905120582(Ω))

Abstract and Applied Analysis 5

Using this last inequality and Holder inequality weconclude our proof since

119906119901

119871119901(Ω)le intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 119862119886 (119906 |119906|119901minus2

119906) = 119862intΩ

119891|119906|119901minus2

119906119889119909

le 119862intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909 le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(38)

If (ℎ1) (ℎ2) and (ℎ4) hold we consider again thefunctions 119906119904 119904 = 1 119903 obtained in correspondence ofthe solution 119906 and of 119892 as in the previous case and of 120576 asin Lemma 41 of [5] In this second case easy computationstogether with (29) give

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901119889119909 (39)

with 1198880 = 1198880(119903 119901)Thus from (33) we deduce that

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

119862s

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

le 1198881

119903

sum

119904=1

119886 (11990610038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904)

(40)

with 119862119904 = 119862119904(119904 119903 ] 120583) and 1198881 = 1198881(119903 119901 ] 120583)Hence by (28) and Holder inequality we obtain

119906119901

119871119901(Ω)le 1198881

119903

sum

119904=1

intΩ

11989110038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904119889119909

le 1199031198881intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909

le 119903119888110038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(41)

This ends the proof in view of (30)

In the later paper [6] estimate (34) has been improveddropping the hypotheses on the boundedness of 119891 and 119906 bymeans of the theorem below

Theorem 8 Assume that hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]2 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (42)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

The proof which is different according to hypothesis (ℎ3)or (ℎ4) is essentially performed into two steps In the firststep we show some regularity results exploiting a technique

introduced by Miranda in [21] Namely we prove that if 119906 isin∘

11988212(Ω) is the solution of (22) with119891 isin 119871

2(Ω)cap119871

infin(Ω) then

the datum 119891 being more regular one also has 119906 isin 119871infin(Ω)

Thus Theorem 7 applies giving that 119906 isin 119871119901(Ω) and satisfies

(34) The second step consists in considering a datum 119891 isin

1198712(Ω) cap 119871

119901(Ω) and then one can conclude by means of some

approximation arguments see also [16]Finally in [6] we prove the main result that is the

claimed 119871119901-bound 119901 gt 1 To this aim a further assumptionon the leading coefficients is required

119886119894119895 = 119886119895119894 119894 119895 = 1 119899 (ℎ0)

Then one has the following

Theorem 9 Assume that hypotheses (ℎ0)ndash(ℎ3) or (ℎ0) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]1 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (43)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof For 119901 ge 2 Theorems 2 and 8 already prove the resultIt remains to show it for 1 lt 119901 lt 2

We assume that hypotheses (ℎ0)ndash(ℎ3) hold true Underhypotheses (ℎ0) (ℎ2) and (ℎ4) a similar argument withsuitable modifications can be used (we refer the reader to [6]for the details)

Let us define the bilinear form

119886lowast(119908 V) = 119886 (V 119908) 119908 V isin

∘

11988212(Ω) (44)

By (ℎ0) one has

119886lowast(119908 V)

= intΩ

(

119899

sum

119894119895=1

(119886119894119895119908119909119894+119887119895119908) V119909119895+(

119899

sum

119894=1

119889119894119908119909119894+119888119908) V)119889119909

(45)

Now consider the problem

119908 isin∘

11988212(Ω)

119886lowast(119908 V) = int

Ω

119892V 119889119909 119892 isin 1198712(Ω) cap 119871

1199011015840

(Ω)

(46)

where since 1 lt 119901 lt 2 one gets 1199011015840 = 119901(119901 minus 1) gt 2As a consequence of Theorem 2 (in the second set of

hypotheses) the solution 119908 of (46) exists and is uniqueFurthermore byTheorem 8 (in the second set of hypotheses)one also has

1199081198711199011015840

(Ω)le 119862

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω) (47)

Hence if we denote by 119906 the solution of

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 1198712(Ω) cap 119871

119901(Ω)

(48)

6 Abstract and Applied Analysis

which exists and is unique in view of Theorem 2 (in the firstset of hypotheses) we obtain

intΩ

119892119906 119889119909 = 119886lowast(119908 119906) = 119886 (119906 119908) = int

Ω

119891119908 119889119909

le10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119908 119871119901

1015840

(Ω)le 119862

10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω)

(49)

Finally taking 119892 = |119906|119901minus1 sign 119906 in (49) we get the claimed

result

4 Non-Variational Problems

In this section we show two applications of ourmain estimate(43)

To this aim let 119901 gt 1 and assume that

Ω has the uniform 11986211-regularity property (ℎ

1015840

0)

Consider then the non-variational differential operator

119871 = minus

119899

sum

119894119895=1

1198861198941198951205972

120597119909119894120597119909119895+

119899

sum

119894=1

119886119894120597

120597119909119894+ 119886 (50)

with the following conditions on the leading coefficients

119886119894119895 = 119886119895119894 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(119886119894119895)119909ℎisin 119872119902120582

119900(Ω) 119894 119895 ℎ = 1 119899 with

119902 gt 2 120582 = 0 for 119899 = 2

119902 isin ] 2 119899 [ 120582 = 119899 minus 119902 for 119899 gt 2

(ℎ1015840

1)

Suppose that the lower order terms are such that

119886119894 isin 119872119903

119900(Ω) 119894 = 1 119899 with

119903 gt 2 if 119901 le 2 119903 = 119901 if 119901 gt 2 for 119899 = 2

119903 ge 119901 119903 ge 119899 with 119903 gt 119901 if 119901 = 119899 for 119899 gt 2

(ℎ1015840

2)

119886 isin 119905(Ω) with

119905 = 119901 for 119899 = 2

119905 ge 119901 119905 ge119899

2 with 119905 gt 119901 if 119901 =

119899

2for 119899 gt 2

ess infΩ

119886 = 1198860 gt 0

(ℎ1015840

3)

In view of Theorem 1 under the assumptions (ℎ10158400)ndash(ℎ10158403) the

operator 119871 1198822119901(Ω) rarr 119871119901(Ω) is bounded

The first application is contained in Theorem 32 andCorollary 33 of [7] (see also [22] where the case 119901 = 2 isconsidered) and reads as follows

Theorem 10 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901(Ω) le 1198881003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901(Ω)

forall119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω) (51)

with 119888= 119888(Ω 119899 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) 120590119902120582

119900[(119886119894119895)119909ℎ

] 120590119903

119900[119886119894]

119905[119886]

1198860)Moreover the problem

119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω)

119871119906 = 119891 119891 isin 119871119901(Ω)

(52)

is uniquely solvable

The nodal point in achieving these results consists in theexistence of the derivatives of the 119886119894119895 Indeed this consentsto rewrite the operator 119871 in divergence form and exploit(43) in order to obtain an estimate as that in (51) but formore regular functions Then one can prove (51) by meansof an approximation argument Estimate (51) immediatelytakes to the solvability of problem (52) via a straightforwardapplication of the method of continuity along a parametersee for instance [23] and by the already known solvabilityof an opportune auxiliary problem

As second application of (43) we obtain in [8] ananalogous of Theorem 10 in a weighted framework Namelywe consider a weight function 120588119904 that is a power of a function120588 of class 1198622(Ω) such that 120588 Ω rarr R+ and

sup119909isinΩ

1003816100381610038161003816120597120572120588 (119909)

1003816100381610038161003816

120588 (119909)lt +infin forall |120572| le 2

lim|119909|rarr+infin

(120588 (119909) +1

120588 (119909)) = +infin

lim|119909|rarr+infin

120588119909(119909) + 120588

119909119909(119909)

120588 (119909)= 0

(53)

For instance one can think of 120588 as the function

120588 (119909) = (1 + |119909|2)119905

119905 isin R 0 (54)

For 119896 isin N0 119901 isin [1 +infin[ and 119904 isin R and given 120588 satisfying(53) we define the weighted Sobolev space 119882119896119901

119904(Ω) as the

space of distributions 119906 onΩ such that

119906119882119896119901

119904 (Ω)= sum

|120572|le119896

10038171003817100381710038171205881199041205971205721199061003817100381710038171003817 119871119901(Ω) lt +infin (55)

endowed with the norm in (55) Furthermore we denote theclosure of 119862infin

∘(Ω) in119882119896119901

119904(Ω) by

∘

119882119896119901

119904(Ω) and put1198820119901

119904(Ω) =

119871119901

119904(Ω)In Theorems 42 and 52 of [8] we showed the following

Theorem 11 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901

119904 (Ω)le 119888

1003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901119904 (Ω)

forall119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

(56)

Abstract and Applied Analysis 7

with 119888 = 119888(Ω 119899 119904 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) ||119886119894||119872119903(Ω) 120590119902120582119900 [(119886119894119895)119909ℎ]120590119903

119900[119886119894]

119905[119886] 1198860)

Moreover the problem

119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

119871119906 = 119891 119891 isin 119871119901

119904(Ω)

(57)

is uniquely solvable

One of the main tools in the proof of Theorem 11 is givenby the existence of a topological isomorphism from119882

119896119901

119904(Ω)

to119882119896119901(Ω) and from∘

119882119896119901

119904(Ω) to

∘

119882119896119901(Ω)This isomorphism

consents to deduce by the non-weighted bound in (51) thecorresponding weighted estimate in (56) taking into accountalso the imbedding results of Theorem 1 The existence anduniqueness of the solution of problem (57) follow then as inthe previous case from a direct application of the method ofcontinuity along a parameter by the solvability of a suitableauxiliary problem

References

[1] G Bottaro and M E Marina ldquoProblema di Dirichlet perequazioni ellittiche di tipo variazionale su insiemi non limitatirdquoBollettino dellrsquoUnione Matematica Italiana Serie 4 vol 8 pp46ndash56 1973

[2] M Transirico and M Troisi ldquoEquazioni ellittiche del secondoordine a coefficienti discontinui e di tipo variazionale in apertinon limitatirdquo Bollettino dellrsquoUnioneMatematica Italiana Serie 7vol 2 no 1 pp 385ndash398 1988

[3] M Transirico M Troisi and A Vitolo ldquoSpaces of Morreytype and elliptic equations in divergence form on unboundeddomainsrdquo Bollettino dellrsquoUnione Matematica Italiana Serie 7vol 9 no 1 pp 153ndash174 1995

[4] S Monsurro and M Transirico ldquoAn 119871119901-estimate for weak

solutions of elliptic equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 376179 15 pages 2012

[5] S Monsurro and M Transirico ldquoDirichlet problem for diver-gence form elliptic equations with discontinuous coefficientsrdquoBoundary Value Problems vol 2012 20 pages 2012

[6] S Monsurro and M Transirico ldquoA priori bounds in 119871119901 for

solutions of elliptic equations in divergence formrdquo Bulletin desSciences Mathematiques In press

[7] S Monsurro and M Transirico ldquoA 1198822119901-estimate for a class

of elliptic operatorsrdquo International Journal of Pure and AppliedMathematics vol 83 no 4 pp 489ndash499 2013

[8] S Monsurro and M Transirico ldquoA weighted 1198822119901-bound

for a class of elliptic operatorsrdquo Journal of Inequalities andApplications vol 2013 article 263 2013

[9] C Miranda ldquoSulle equazioni ellittiche del secondo ordine ditipo non variazionale a coefficienti discontinuirdquo Annali diMatematica Pura ed Applicata vol 63 no 1 pp 353ndash386 1963

[10] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines nonbornesrdquo Atti della Accademia Nazionale dei Lincei RendicontiClasse di Scienze Fisiche Matematiche e Naturali Serie VIII vol78 no 5 pp 205ndash212 1985

[11] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines

non bornes IIrdquo Atti della Accademia Nazionale dei LinceiRendiconti Classe di Scienze Fisiche Matematiche e NaturaliSerie VIII vol 79 no 6 pp 178ndash183 1985

[12] P Donato and D Giachetti ldquoQuasilinear elliptic equations withquadratic growth in unbounded domainsrdquo Nonlinear AnalysisTheoryMethods ampApplications vol 10 no 8 pp 791ndash804 1986

[13] L Caso P Cavaliere and M Transirico ldquoAn existence resultfor elliptic equations with VMO-coefficientsrdquo Journal of Mathe-matical Analysis and Applications vol 325 no 2 pp 1095ndash11022007

[14] S Boccia S Monsurro and M Transirico ldquoElliptic equationsin weighted Sobolev spaces on unbounded domainsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2008 Article ID 582435 12 pages 2008

[15] S Boccia S Monsurro and M Transirico ldquoSolvability of theDirichlet problem for elliptic equations in weighted Sobolevspaces on unbounded domainsrdquo Boundary Value Problems vol2008 Article ID 901503 13 pages 2008

[16] L Caso R DrsquoAmbrosio and S Monsurro ldquoSome remarks onspaces of Morrey typerdquo Abstract and Applied Analysis vol 2010Article ID 242079 22 pages 2010

[17] P Cavaliere M Longobardi and A Vitolo ldquoImbedding esti-mates and elliptic equations with discontinuous coefficients inunbounded domainsrdquo LeMatematiche vol 51 no 1 pp 87ndash1041996

[18] G Stampacchia ldquoLe probleme de Dirichlet pour les equationselliptiques du second ordre a coefficients discontinusrdquo Annalesde lrsquoInstitut Fourier vol 15 pp 189ndash258 1965

[19] G Stampacchia Equations Elliptiques du Second Ordre a Coef-ficients Discontinus Seminaire de Mathematiques Superieuresno 16 (Ete 1965) Les Presses de lrsquoUniversite de MontrealMontreal Canada 1966

[20] L Caso P Cavaliere and M Transirico ldquoSolvability of theDirichlet problem in1198822119901 for elliptic equations with discontin-uous coefficients in unbounded domainsrdquo Le Matematiche vol57 no 2 pp 287ndash302 2002

[21] C Miranda ldquoAlcune osservazioni sulla maggiorazione in 119871]119901

delle soluzioni deboli delle equazioni ellittiche del secondoordinerdquoAnnali diMatematica Pura ed Applicata vol 61 pp 151ndash169 1963

[22] S Monsurro M Salvato and M Transirico ldquo11988222 a prioribounds for a class of elliptic operatorsrdquo International Journalof Differential Equations vol 2011 Article ID 572824 17 pages2011

[23] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 Springer Berlin Germany2nd edition 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

Abstract and Applied Analysis 3

Theorem 1 If 119892 isin 119872119902120582(Ω) with 119902 gt 2 and 120582 = 0 if 119899 = 2

and 119902 isin ]2 119899] and 120582 = 119899 minus 119902 if 119899 gt 2 then the operator in(17) is bounded from

∘

11988212(Ω) to 1198712(Ω) Moreover there exists

a constant 119862 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 1198712(Ω) le 119862

10038171003817100381710038171198921003817100381710038171003817119872119902120582(Ω)11990611988212(Ω) forall119906 isin

∘

11988212(Ω) (18)

with 119862 = 119862(119899 119902)Let 119901 gt 1 and 119903 119905 isin [119901 +infin[ If Ω is an open subset of R119899

having the cone property and 119892 isin 119872119903(Ω) with 119903 gt 119901 if 119901 = 119899

then the operator in (17) is bounded from 1198821119901(Ω) to 119871119901(Ω)

Moreover there exists a constant 119888 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 119871119901(Ω) le 119888

10038171003817100381710038171198921003817100381710038171003817119872119903(Ω)1199061198821119901(Ω) forall119906 isin 119882

1119901(Ω) (19)

with 119888 = 119888 (Ω 119899 119901 119903)If 119892 isin 119872

119905(Ω) with 119905 gt 119901 if 119901 = 1198992 then the operator in

(17) is bounded from1198822119901(Ω) to 119871119901(Ω) Moreover there exists

a constant 1198881015840 isin R+ such that

10038171003817100381710038171198921199061003817100381710038171003817 119871119901(Ω) le 119888

101584010038171003817100381710038171198921003817100381710038171003817119872119905(Ω)1199061198822119901(Ω) forall119906 isin 119882

2119901(Ω) (20)

with 1198881015840 = 1198881015840(Ω 119899 119901 119905)

3 The Variational Problem

Consider in an unbounded open subset Ω of R119899 119899 ge 2 thesecond-order linear differential operator in divergence form

119871 = minus

119899

sum

119894119895=1

120597

120597119909119895(119886119894119895

120597

120597119909119894+ 119889119895) +

119899

sum

119894=1

119887119894120597

120597119909119894+ 119888 (21)

Assume that the leading coefficients satisfy the hypotheses

119886119894119895 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(ℎ1)

For the lower order terms coefficients suppose that

119887119894 119889119894 isin 1198722119905120582

119900(Ω) 119894 = 1 119899

119888 isin 119872119905120582(Ω)

with 119905 gt 1 and 120582 = 0 if 119899 = 2

with 119905 isin ]1119899

2] and 120582 = 119899 minus 2119905 if 119899 gt 2

(ℎ2)

Furthermore let one of the following sign assumptions holdtrue

119888 minus

119899

sum

119894=1

(119889119894)119909119894ge 120583 (ℎ3)

or

119888 minus

119899

sum

119894=1

(119887119894)119909119894ge 120583 (ℎ4)

in the distributional sense onΩ with 120583 positive constant

We are interested in the study of the Dirichlet problem

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 119882minus12

(Ω)

(22)

(ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2) and (ℎ4) being satisfiedIt is natural to associate to 119871 the bilinear form

119886 (119906 V) = intΩ

(

119899

sum

119894119895=1

(119886119894119895119906119909119894+ 119889119895119906) V119909119895

+(

119899

sum

119894=1

119887119894119906119909119894+ 119888119906) V)119889119909

(23)

119906 V isin∘

11988212(Ω) and observe that in view of Theorem 1 the

form 119886 is continuous on∘

11988212(Ω) times

∘

11988212(Ω) and so the

operator 119871 ∘

11988212(Ω) rarr 119882

minus12(Ω) is continuous too

Let us start collecting some preliminary results concern-ing the existence and uniqueness of the solution of problem(22) as well as some a priori estimates For the case whereassumptions (ℎ1)ndash(ℎ3) are taken into account and for 119899 = 2we refer to [2] while for 119899 ge 3 details can be found in [3]If (ℎ1) (ℎ2) and (ℎ4) hold true the results are proved in themore recent [5]

Theorem 2 Under hypotheses (ℎ1)ndash(ℎ3) (or (ℎ1) (ℎ2) and(ℎ4)) problem (22) is uniquely solvable and its solution 119906

satisfies the estimate

11990611988212(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817119882minus12(Ω) (24)

where 119862 is a constant depending on 119899 119905 ] 120583 ||119887119894 minus 119889119894||1198722119905120582(Ω)119894 = 1 119899

The next step in our analysis is to achieve an 119871119901-estimate119901 gt 2 for the solution of (22) (see Theorem 8) This requiressome additional hypotheses on the regularity of the set andon the datum 119891 and some preparatory results that essentiallyrely on the introduction of certain auxiliary functions119906119904 usedfor the first time by Bottaro and Marina in [1] and employedin the framework of Morrey type spaces in [3] Let us givetheir definition and recall some useful properties

Let ℎ isin R+ cup +infin and 119896 isin R with 0 le 119896 le ℎ For each119905 isin R we set

119866119896ℎ (119905) =

119905 minus 119896 if 119905 gt 119896

0 if minus 119896 le 119905 le 119896 if ℎ = +infin

119905 + 119896 if 119905 lt minus119896

119866119896ℎ (119905) = 119866119896infin (119905) minus 119866ℎinfin (119905) if ℎ isin R+

(25)

Lemma 3 Let 119892 isin 119872119902120582

119900(Ω) 119906 isin

∘

11988212(Ω) and 120576 isin R+ Then

there exist 119903 isin N and 1198961 119896119903 isin R with 0 = 119896119903 lt 119896119903minus1 lt

sdot sdot sdot lt 1198961 lt 1198960 = +infin such that set

119906119904 = 119866119896119904119896119904minus1(119906) 119904 = 1 119903 (26)

4 Abstract and Applied Analysis

one has 1199061 119906119903 isin∘

11988212(Ω) and

10038171003817100381710038171003817119892120594supp (119906119904)119909

10038171003817100381710038171003817119872119902120582(Ω)le 120576 119904 = 1 119903 (27)

10038161003816100381610038161199061199041003816100381610038161003816 le |119906| 119904 = 1 119903 (28)

1199061 + sdot sdot sdot + 119906119903 = 119906 (29)

119903 le 119888 (30)

with 119888 = 119888 (120576 119902 119892119872119902120582(Ω)

) positive constant

In order to prove a fundamental preliminary estimateobtained for 119901 gt 2 (seeTheorem 7) we need to take productsinvolving the above defined functions 119906119904 as test functions inthe variational formulation of our problem (23) To be moreprecise in the first set of hypotheses ((ℎ1)ndash(ℎ3)) the testfunctions needed are |119906|119901minus2119906119904 The following result ensuresthat these functions effectively belong to

∘

11988212(Ω)

Lemma 4 If Ω has the uniform 1198621-regularity property then

for every 119906 isin∘

11988212(Ω) cap 119871

infin(Ω) and for any 119901 isin ]2 +infin[ one

has

|119906|119901minus2

119906119904 isin∘

11988212(Ω) 119904 = 1 119903 (31)

Lemma 4 whose rather technical proof can be found in [4]is a generalization of a known result by Stampacchia (see[18] or [19] for details) obtained within the framework ofthe generalization of the study of certain elliptic equationsin divergence form with discontinuous coefficients on abounded open subset of R119899 to some problems arisingfor harmonic or subharmonic functions in the theory ofpotential

Once achieved (31) always in [4] we could prove thenext lemma Let 119906119904 be the functions of Lemma 3 obtainedin correspondence of a given 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of

119892 = sum119899

119894=1|119887119894 minus 119889119894| and of a positive real number 120576 specified

in the proof of Lemma 41 of [4] One has the following

Lemma 5 Let 119886 be the bilinear form defined in (23) If Ωhas the uniform119862

1-regularity property under hypotheses (ℎ1)ndash(ℎ3) there exists a constant 119862 isin R+ such that

intΩ

|119906|119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(32)

where 119862 depends on 119904 ] 120583

If we consider the second set of hypotheses ((ℎ1) (ℎ2)and (ℎ4)) the test functions required in (23) are the prod-ucts |119906119904|

119901minus2119906119904 obtained in correspondence of a fixed 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of 119892 = sum

119899

119894=1|119889119894 minus 119887119894| and of a positive real

number 120576 specified in the proof of Lemma 41 of [5] In thislast case and if Ω has the uniform 119862

1-regularity property aresult of [20] applies giving that |119906119904|

119901minus2119906119904 isin

∘

11988212(Ω) for any

119901 gt 2 119904 = 1 119903 Hence in [5] we could show the result

Lemma 6 Let 119886 be the bilinear form in (23) If Ω has theuniform 119862

1-regularity property under hypotheses (ℎ1) (ℎ2)and (ℎ4) there exists a constant 119862 isin R+ such that

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(33)

where 119862 depends on 119904 119903 ] 120583

The two lemmas just stated put us in a position to provethe following preliminary 119871

119901-a priori estimate 119901 gt 2 inboth sets of hypotheses see also [4 5] We stress that here werequire that both the datum119891 and the solution119906 are bounded

Theorem 7 Under hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2) and(ℎ4) and if Ω has the uniform 119862

1-regularity property 119891 is in1198712(Ω) cap 119871

infin(Ω) and the solution 119906 of (22) is in

∘

11988212(Ω) cap

119871infin(Ω) then 119906 isin 119871119901(Ω) and

119906 119871119901(Ω) le C10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) forall119901 isin ]2 +infin[ (34)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof Fix 119901 isin ]2 +infin[ We provide two different proofs inthe cases that hypotheses (ℎ3) or (ℎ4) hold true

Let (ℎ1)ndash(ℎ3) be satisfied We consider the functions 119906119904119904 = 1 119903 obtained in correspondence of the solution 119906 andof 119892 = sum

119899

119894=1|1198891 minus 119887119894| and of 120576 as in Lemma 41 of [4] In view

of (29) we get

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880intΩ

|119906|119901minus2119903

sum

119904=1

((119906119904)2

119909+ 1199062

119904) 119889119909

(35)

with 1198880 = 1198880(119903)Hence (32) entails that

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880

119903

sum

119904=1

119862119904

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

le 119862

119903

sum

119904=1

119886 (119906 |119906|119901minus2

119906119904)

(36)

with 119862119904 = 119862119904(119904 ] 120583) and 119862 = 119862(119903 ] 120583)From the linearity of 119886 (29) and (30) we have then

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909 le 119862119886 (119906 |119906|

119901minus2119906) (37)

with 119862 = 119862(119899 119905 119901 ] 120583 ||119887119894 minus 119889119894||1198722119905120582(Ω))

Abstract and Applied Analysis 5

Using this last inequality and Holder inequality weconclude our proof since

119906119901

119871119901(Ω)le intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 119862119886 (119906 |119906|119901minus2

119906) = 119862intΩ

119891|119906|119901minus2

119906119889119909

le 119862intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909 le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(38)

If (ℎ1) (ℎ2) and (ℎ4) hold we consider again thefunctions 119906119904 119904 = 1 119903 obtained in correspondence ofthe solution 119906 and of 119892 as in the previous case and of 120576 asin Lemma 41 of [5] In this second case easy computationstogether with (29) give

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901119889119909 (39)

with 1198880 = 1198880(119903 119901)Thus from (33) we deduce that

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

119862s

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

le 1198881

119903

sum

119904=1

119886 (11990610038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904)

(40)

with 119862119904 = 119862119904(119904 119903 ] 120583) and 1198881 = 1198881(119903 119901 ] 120583)Hence by (28) and Holder inequality we obtain

119906119901

119871119901(Ω)le 1198881

119903

sum

119904=1

intΩ

11989110038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904119889119909

le 1199031198881intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909

le 119903119888110038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(41)

This ends the proof in view of (30)

In the later paper [6] estimate (34) has been improveddropping the hypotheses on the boundedness of 119891 and 119906 bymeans of the theorem below

Theorem 8 Assume that hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]2 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (42)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

The proof which is different according to hypothesis (ℎ3)or (ℎ4) is essentially performed into two steps In the firststep we show some regularity results exploiting a technique

introduced by Miranda in [21] Namely we prove that if 119906 isin∘

11988212(Ω) is the solution of (22) with119891 isin 119871

2(Ω)cap119871

infin(Ω) then

the datum 119891 being more regular one also has 119906 isin 119871infin(Ω)

Thus Theorem 7 applies giving that 119906 isin 119871119901(Ω) and satisfies

(34) The second step consists in considering a datum 119891 isin

1198712(Ω) cap 119871

119901(Ω) and then one can conclude by means of some

approximation arguments see also [16]Finally in [6] we prove the main result that is the

claimed 119871119901-bound 119901 gt 1 To this aim a further assumptionon the leading coefficients is required

119886119894119895 = 119886119895119894 119894 119895 = 1 119899 (ℎ0)

Then one has the following

Theorem 9 Assume that hypotheses (ℎ0)ndash(ℎ3) or (ℎ0) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]1 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (43)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof For 119901 ge 2 Theorems 2 and 8 already prove the resultIt remains to show it for 1 lt 119901 lt 2

We assume that hypotheses (ℎ0)ndash(ℎ3) hold true Underhypotheses (ℎ0) (ℎ2) and (ℎ4) a similar argument withsuitable modifications can be used (we refer the reader to [6]for the details)

Let us define the bilinear form

119886lowast(119908 V) = 119886 (V 119908) 119908 V isin

∘

11988212(Ω) (44)

By (ℎ0) one has

119886lowast(119908 V)

= intΩ

(

119899

sum

119894119895=1

(119886119894119895119908119909119894+119887119895119908) V119909119895+(

119899

sum

119894=1

119889119894119908119909119894+119888119908) V)119889119909

(45)

Now consider the problem

119908 isin∘

11988212(Ω)

119886lowast(119908 V) = int

Ω

119892V 119889119909 119892 isin 1198712(Ω) cap 119871

1199011015840

(Ω)

(46)

where since 1 lt 119901 lt 2 one gets 1199011015840 = 119901(119901 minus 1) gt 2As a consequence of Theorem 2 (in the second set of

hypotheses) the solution 119908 of (46) exists and is uniqueFurthermore byTheorem 8 (in the second set of hypotheses)one also has

1199081198711199011015840

(Ω)le 119862

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω) (47)

Hence if we denote by 119906 the solution of

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 1198712(Ω) cap 119871

119901(Ω)

(48)

6 Abstract and Applied Analysis

which exists and is unique in view of Theorem 2 (in the firstset of hypotheses) we obtain

intΩ

119892119906 119889119909 = 119886lowast(119908 119906) = 119886 (119906 119908) = int

Ω

119891119908 119889119909

le10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119908 119871119901

1015840

(Ω)le 119862

10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω)

(49)

Finally taking 119892 = |119906|119901minus1 sign 119906 in (49) we get the claimed

result

4 Non-Variational Problems

In this section we show two applications of ourmain estimate(43)

To this aim let 119901 gt 1 and assume that

Ω has the uniform 11986211-regularity property (ℎ

1015840

0)

Consider then the non-variational differential operator

119871 = minus

119899

sum

119894119895=1

1198861198941198951205972

120597119909119894120597119909119895+

119899

sum

119894=1

119886119894120597

120597119909119894+ 119886 (50)

with the following conditions on the leading coefficients

119886119894119895 = 119886119895119894 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(119886119894119895)119909ℎisin 119872119902120582

119900(Ω) 119894 119895 ℎ = 1 119899 with

119902 gt 2 120582 = 0 for 119899 = 2

119902 isin ] 2 119899 [ 120582 = 119899 minus 119902 for 119899 gt 2

(ℎ1015840

1)

Suppose that the lower order terms are such that

119886119894 isin 119872119903

119900(Ω) 119894 = 1 119899 with

119903 gt 2 if 119901 le 2 119903 = 119901 if 119901 gt 2 for 119899 = 2

119903 ge 119901 119903 ge 119899 with 119903 gt 119901 if 119901 = 119899 for 119899 gt 2

(ℎ1015840

2)

119886 isin 119905(Ω) with

119905 = 119901 for 119899 = 2

119905 ge 119901 119905 ge119899

2 with 119905 gt 119901 if 119901 =

119899

2for 119899 gt 2

ess infΩ

119886 = 1198860 gt 0

(ℎ1015840

3)

In view of Theorem 1 under the assumptions (ℎ10158400)ndash(ℎ10158403) the

operator 119871 1198822119901(Ω) rarr 119871119901(Ω) is bounded

The first application is contained in Theorem 32 andCorollary 33 of [7] (see also [22] where the case 119901 = 2 isconsidered) and reads as follows

Theorem 10 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901(Ω) le 1198881003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901(Ω)

forall119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω) (51)

with 119888= 119888(Ω 119899 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) 120590119902120582

119900[(119886119894119895)119909ℎ

] 120590119903

119900[119886119894]

119905[119886]

1198860)Moreover the problem

119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω)

119871119906 = 119891 119891 isin 119871119901(Ω)

(52)

is uniquely solvable

The nodal point in achieving these results consists in theexistence of the derivatives of the 119886119894119895 Indeed this consentsto rewrite the operator 119871 in divergence form and exploit(43) in order to obtain an estimate as that in (51) but formore regular functions Then one can prove (51) by meansof an approximation argument Estimate (51) immediatelytakes to the solvability of problem (52) via a straightforwardapplication of the method of continuity along a parametersee for instance [23] and by the already known solvabilityof an opportune auxiliary problem

As second application of (43) we obtain in [8] ananalogous of Theorem 10 in a weighted framework Namelywe consider a weight function 120588119904 that is a power of a function120588 of class 1198622(Ω) such that 120588 Ω rarr R+ and

sup119909isinΩ

1003816100381610038161003816120597120572120588 (119909)

1003816100381610038161003816

120588 (119909)lt +infin forall |120572| le 2

lim|119909|rarr+infin

(120588 (119909) +1

120588 (119909)) = +infin

lim|119909|rarr+infin

120588119909(119909) + 120588

119909119909(119909)

120588 (119909)= 0

(53)

For instance one can think of 120588 as the function

120588 (119909) = (1 + |119909|2)119905

119905 isin R 0 (54)

For 119896 isin N0 119901 isin [1 +infin[ and 119904 isin R and given 120588 satisfying(53) we define the weighted Sobolev space 119882119896119901

119904(Ω) as the

space of distributions 119906 onΩ such that

119906119882119896119901

119904 (Ω)= sum

|120572|le119896

10038171003817100381710038171205881199041205971205721199061003817100381710038171003817 119871119901(Ω) lt +infin (55)

endowed with the norm in (55) Furthermore we denote theclosure of 119862infin

∘(Ω) in119882119896119901

119904(Ω) by

∘

119882119896119901

119904(Ω) and put1198820119901

119904(Ω) =

119871119901

119904(Ω)In Theorems 42 and 52 of [8] we showed the following

Theorem 11 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901

119904 (Ω)le 119888

1003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901119904 (Ω)

forall119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

(56)

Abstract and Applied Analysis 7

with 119888 = 119888(Ω 119899 119904 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) ||119886119894||119872119903(Ω) 120590119902120582119900 [(119886119894119895)119909ℎ]120590119903

119900[119886119894]

119905[119886] 1198860)

Moreover the problem

119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

119871119906 = 119891 119891 isin 119871119901

119904(Ω)

(57)

is uniquely solvable

One of the main tools in the proof of Theorem 11 is givenby the existence of a topological isomorphism from119882

119896119901

119904(Ω)

to119882119896119901(Ω) and from∘

119882119896119901

119904(Ω) to

∘

119882119896119901(Ω)This isomorphism

consents to deduce by the non-weighted bound in (51) thecorresponding weighted estimate in (56) taking into accountalso the imbedding results of Theorem 1 The existence anduniqueness of the solution of problem (57) follow then as inthe previous case from a direct application of the method ofcontinuity along a parameter by the solvability of a suitableauxiliary problem

References

[1] G Bottaro and M E Marina ldquoProblema di Dirichlet perequazioni ellittiche di tipo variazionale su insiemi non limitatirdquoBollettino dellrsquoUnione Matematica Italiana Serie 4 vol 8 pp46ndash56 1973

[2] M Transirico and M Troisi ldquoEquazioni ellittiche del secondoordine a coefficienti discontinui e di tipo variazionale in apertinon limitatirdquo Bollettino dellrsquoUnioneMatematica Italiana Serie 7vol 2 no 1 pp 385ndash398 1988

[3] M Transirico M Troisi and A Vitolo ldquoSpaces of Morreytype and elliptic equations in divergence form on unboundeddomainsrdquo Bollettino dellrsquoUnione Matematica Italiana Serie 7vol 9 no 1 pp 153ndash174 1995

[4] S Monsurro and M Transirico ldquoAn 119871119901-estimate for weak

solutions of elliptic equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 376179 15 pages 2012

[5] S Monsurro and M Transirico ldquoDirichlet problem for diver-gence form elliptic equations with discontinuous coefficientsrdquoBoundary Value Problems vol 2012 20 pages 2012

[6] S Monsurro and M Transirico ldquoA priori bounds in 119871119901 for

solutions of elliptic equations in divergence formrdquo Bulletin desSciences Mathematiques In press

[7] S Monsurro and M Transirico ldquoA 1198822119901-estimate for a class

of elliptic operatorsrdquo International Journal of Pure and AppliedMathematics vol 83 no 4 pp 489ndash499 2013

[8] S Monsurro and M Transirico ldquoA weighted 1198822119901-bound

for a class of elliptic operatorsrdquo Journal of Inequalities andApplications vol 2013 article 263 2013

[9] C Miranda ldquoSulle equazioni ellittiche del secondo ordine ditipo non variazionale a coefficienti discontinuirdquo Annali diMatematica Pura ed Applicata vol 63 no 1 pp 353ndash386 1963

[10] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines nonbornesrdquo Atti della Accademia Nazionale dei Lincei RendicontiClasse di Scienze Fisiche Matematiche e Naturali Serie VIII vol78 no 5 pp 205ndash212 1985

[11] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines

non bornes IIrdquo Atti della Accademia Nazionale dei LinceiRendiconti Classe di Scienze Fisiche Matematiche e NaturaliSerie VIII vol 79 no 6 pp 178ndash183 1985

[12] P Donato and D Giachetti ldquoQuasilinear elliptic equations withquadratic growth in unbounded domainsrdquo Nonlinear AnalysisTheoryMethods ampApplications vol 10 no 8 pp 791ndash804 1986

[13] L Caso P Cavaliere and M Transirico ldquoAn existence resultfor elliptic equations with VMO-coefficientsrdquo Journal of Mathe-matical Analysis and Applications vol 325 no 2 pp 1095ndash11022007

[14] S Boccia S Monsurro and M Transirico ldquoElliptic equationsin weighted Sobolev spaces on unbounded domainsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2008 Article ID 582435 12 pages 2008

[15] S Boccia S Monsurro and M Transirico ldquoSolvability of theDirichlet problem for elliptic equations in weighted Sobolevspaces on unbounded domainsrdquo Boundary Value Problems vol2008 Article ID 901503 13 pages 2008

[16] L Caso R DrsquoAmbrosio and S Monsurro ldquoSome remarks onspaces of Morrey typerdquo Abstract and Applied Analysis vol 2010Article ID 242079 22 pages 2010

[17] P Cavaliere M Longobardi and A Vitolo ldquoImbedding esti-mates and elliptic equations with discontinuous coefficients inunbounded domainsrdquo LeMatematiche vol 51 no 1 pp 87ndash1041996

[18] G Stampacchia ldquoLe probleme de Dirichlet pour les equationselliptiques du second ordre a coefficients discontinusrdquo Annalesde lrsquoInstitut Fourier vol 15 pp 189ndash258 1965

[19] G Stampacchia Equations Elliptiques du Second Ordre a Coef-ficients Discontinus Seminaire de Mathematiques Superieuresno 16 (Ete 1965) Les Presses de lrsquoUniversite de MontrealMontreal Canada 1966

[20] L Caso P Cavaliere and M Transirico ldquoSolvability of theDirichlet problem in1198822119901 for elliptic equations with discontin-uous coefficients in unbounded domainsrdquo Le Matematiche vol57 no 2 pp 287ndash302 2002

[21] C Miranda ldquoAlcune osservazioni sulla maggiorazione in 119871]119901

delle soluzioni deboli delle equazioni ellittiche del secondoordinerdquoAnnali diMatematica Pura ed Applicata vol 61 pp 151ndash169 1963

[22] S Monsurro M Salvato and M Transirico ldquo11988222 a prioribounds for a class of elliptic operatorsrdquo International Journalof Differential Equations vol 2011 Article ID 572824 17 pages2011

[23] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 Springer Berlin Germany2nd edition 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

4 Abstract and Applied Analysis

one has 1199061 119906119903 isin∘

11988212(Ω) and

10038171003817100381710038171003817119892120594supp (119906119904)119909

10038171003817100381710038171003817119872119902120582(Ω)le 120576 119904 = 1 119903 (27)

10038161003816100381610038161199061199041003816100381610038161003816 le |119906| 119904 = 1 119903 (28)

1199061 + sdot sdot sdot + 119906119903 = 119906 (29)

119903 le 119888 (30)

with 119888 = 119888 (120576 119902 119892119872119902120582(Ω)

) positive constant

In order to prove a fundamental preliminary estimateobtained for 119901 gt 2 (seeTheorem 7) we need to take productsinvolving the above defined functions 119906119904 as test functions inthe variational formulation of our problem (23) To be moreprecise in the first set of hypotheses ((ℎ1)ndash(ℎ3)) the testfunctions needed are |119906|119901minus2119906119904 The following result ensuresthat these functions effectively belong to

∘

11988212(Ω)

Lemma 4 If Ω has the uniform 1198621-regularity property then

for every 119906 isin∘

11988212(Ω) cap 119871

infin(Ω) and for any 119901 isin ]2 +infin[ one

has

|119906|119901minus2

119906119904 isin∘

11988212(Ω) 119904 = 1 119903 (31)

Lemma 4 whose rather technical proof can be found in [4]is a generalization of a known result by Stampacchia (see[18] or [19] for details) obtained within the framework ofthe generalization of the study of certain elliptic equationsin divergence form with discontinuous coefficients on abounded open subset of R119899 to some problems arisingfor harmonic or subharmonic functions in the theory ofpotential

Once achieved (31) always in [4] we could prove thenext lemma Let 119906119904 be the functions of Lemma 3 obtainedin correspondence of a given 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of

119892 = sum119899

119894=1|119887119894 minus 119889119894| and of a positive real number 120576 specified

in the proof of Lemma 41 of [4] One has the following

Lemma 5 Let 119886 be the bilinear form defined in (23) If Ωhas the uniform119862

1-regularity property under hypotheses (ℎ1)ndash(ℎ3) there exists a constant 119862 isin R+ such that

intΩ

|119906|119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(32)

where 119862 depends on 119904 ] 120583

If we consider the second set of hypotheses ((ℎ1) (ℎ2)and (ℎ4)) the test functions required in (23) are the prod-ucts |119906119904|

119901minus2119906119904 obtained in correspondence of a fixed 119906 isin

∘

11988212(Ω) cap 119871

infin(Ω) of 119892 = sum

119899

119894=1|119889119894 minus 119887119894| and of a positive real

number 120576 specified in the proof of Lemma 41 of [5] In thislast case and if Ω has the uniform 119862

1-regularity property aresult of [20] applies giving that |119906119904|

119901minus2119906119904 isin

∘

11988212(Ω) for any

119901 gt 2 119904 = 1 119903 Hence in [5] we could show the result

Lemma 6 Let 119886 be the bilinear form in (23) If Ω has theuniform 119862

1-regularity property under hypotheses (ℎ1) (ℎ2)and (ℎ4) there exists a constant 119862 isin R+ such that

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901minus2

((119906119904)2

119909+ 1199062

119904) 119889119909 le 119862

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

119904 = 1 119903 forall119901 isin ]2 +infin[

(33)

where 119862 depends on 119904 119903 ] 120583

The two lemmas just stated put us in a position to provethe following preliminary 119871

119901-a priori estimate 119901 gt 2 inboth sets of hypotheses see also [4 5] We stress that here werequire that both the datum119891 and the solution119906 are bounded

Theorem 7 Under hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2) and(ℎ4) and if Ω has the uniform 119862

1-regularity property 119891 is in1198712(Ω) cap 119871

infin(Ω) and the solution 119906 of (22) is in

∘

11988212(Ω) cap

119871infin(Ω) then 119906 isin 119871119901(Ω) and

119906 119871119901(Ω) le C10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) forall119901 isin ]2 +infin[ (34)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof Fix 119901 isin ]2 +infin[ We provide two different proofs inthe cases that hypotheses (ℎ3) or (ℎ4) hold true

Let (ℎ1)ndash(ℎ3) be satisfied We consider the functions 119906119904119904 = 1 119903 obtained in correspondence of the solution 119906 andof 119892 = sum

119899

119894=1|1198891 minus 119887119894| and of 120576 as in Lemma 41 of [4] In view

of (29) we get

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880intΩ

|119906|119901minus2119903

sum

119904=1

((119906119904)2

119909+ 1199062

119904) 119889119909

(35)

with 1198880 = 1198880(119903)Hence (32) entails that

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 1198880

119903

sum

119904=1

119862119904

119904

sum

ℎ=1

119886 (119906 |119906|119901minus2

119906ℎ)

le 119862

119903

sum

119904=1

119886 (119906 |119906|119901minus2

119906119904)

(36)

with 119862119904 = 119862119904(119904 ] 120583) and 119862 = 119862(119903 ] 120583)From the linearity of 119886 (29) and (30) we have then

intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909 le 119862119886 (119906 |119906|

119901minus2119906) (37)

with 119862 = 119862(119899 119905 119901 ] 120583 ||119887119894 minus 119889119894||1198722119905120582(Ω))

Abstract and Applied Analysis 5

Using this last inequality and Holder inequality weconclude our proof since

119906119901

119871119901(Ω)le intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 119862119886 (119906 |119906|119901minus2

119906) = 119862intΩ

119891|119906|119901minus2

119906119889119909

le 119862intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909 le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(38)

If (ℎ1) (ℎ2) and (ℎ4) hold we consider again thefunctions 119906119904 119904 = 1 119903 obtained in correspondence ofthe solution 119906 and of 119892 as in the previous case and of 120576 asin Lemma 41 of [5] In this second case easy computationstogether with (29) give

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901119889119909 (39)

with 1198880 = 1198880(119903 119901)Thus from (33) we deduce that

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

119862s

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

le 1198881

119903

sum

119904=1

119886 (11990610038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904)

(40)

with 119862119904 = 119862119904(119904 119903 ] 120583) and 1198881 = 1198881(119903 119901 ] 120583)Hence by (28) and Holder inequality we obtain

119906119901

119871119901(Ω)le 1198881

119903

sum

119904=1

intΩ

11989110038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904119889119909

le 1199031198881intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909

le 119903119888110038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(41)

This ends the proof in view of (30)

In the later paper [6] estimate (34) has been improveddropping the hypotheses on the boundedness of 119891 and 119906 bymeans of the theorem below

Theorem 8 Assume that hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]2 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (42)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

The proof which is different according to hypothesis (ℎ3)or (ℎ4) is essentially performed into two steps In the firststep we show some regularity results exploiting a technique

introduced by Miranda in [21] Namely we prove that if 119906 isin∘

11988212(Ω) is the solution of (22) with119891 isin 119871

2(Ω)cap119871

infin(Ω) then

the datum 119891 being more regular one also has 119906 isin 119871infin(Ω)

Thus Theorem 7 applies giving that 119906 isin 119871119901(Ω) and satisfies

(34) The second step consists in considering a datum 119891 isin

1198712(Ω) cap 119871

119901(Ω) and then one can conclude by means of some

approximation arguments see also [16]Finally in [6] we prove the main result that is the

claimed 119871119901-bound 119901 gt 1 To this aim a further assumptionon the leading coefficients is required

119886119894119895 = 119886119895119894 119894 119895 = 1 119899 (ℎ0)

Then one has the following

Theorem 9 Assume that hypotheses (ℎ0)ndash(ℎ3) or (ℎ0) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]1 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (43)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof For 119901 ge 2 Theorems 2 and 8 already prove the resultIt remains to show it for 1 lt 119901 lt 2

We assume that hypotheses (ℎ0)ndash(ℎ3) hold true Underhypotheses (ℎ0) (ℎ2) and (ℎ4) a similar argument withsuitable modifications can be used (we refer the reader to [6]for the details)

Let us define the bilinear form

119886lowast(119908 V) = 119886 (V 119908) 119908 V isin

∘

11988212(Ω) (44)

By (ℎ0) one has

119886lowast(119908 V)

= intΩ

(

119899

sum

119894119895=1

(119886119894119895119908119909119894+119887119895119908) V119909119895+(

119899

sum

119894=1

119889119894119908119909119894+119888119908) V)119889119909

(45)

Now consider the problem

119908 isin∘

11988212(Ω)

119886lowast(119908 V) = int

Ω

119892V 119889119909 119892 isin 1198712(Ω) cap 119871

1199011015840

(Ω)

(46)

where since 1 lt 119901 lt 2 one gets 1199011015840 = 119901(119901 minus 1) gt 2As a consequence of Theorem 2 (in the second set of

hypotheses) the solution 119908 of (46) exists and is uniqueFurthermore byTheorem 8 (in the second set of hypotheses)one also has

1199081198711199011015840

(Ω)le 119862

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω) (47)

Hence if we denote by 119906 the solution of

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 1198712(Ω) cap 119871

119901(Ω)

(48)

6 Abstract and Applied Analysis

which exists and is unique in view of Theorem 2 (in the firstset of hypotheses) we obtain

intΩ

119892119906 119889119909 = 119886lowast(119908 119906) = 119886 (119906 119908) = int

Ω

119891119908 119889119909

le10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119908 119871119901

1015840

(Ω)le 119862

10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω)

(49)

Finally taking 119892 = |119906|119901minus1 sign 119906 in (49) we get the claimed

result

4 Non-Variational Problems

In this section we show two applications of ourmain estimate(43)

To this aim let 119901 gt 1 and assume that

Ω has the uniform 11986211-regularity property (ℎ

1015840

0)

Consider then the non-variational differential operator

119871 = minus

119899

sum

119894119895=1

1198861198941198951205972

120597119909119894120597119909119895+

119899

sum

119894=1

119886119894120597

120597119909119894+ 119886 (50)

with the following conditions on the leading coefficients

119886119894119895 = 119886119895119894 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(119886119894119895)119909ℎisin 119872119902120582

119900(Ω) 119894 119895 ℎ = 1 119899 with

119902 gt 2 120582 = 0 for 119899 = 2

119902 isin ] 2 119899 [ 120582 = 119899 minus 119902 for 119899 gt 2

(ℎ1015840

1)

Suppose that the lower order terms are such that

119886119894 isin 119872119903

119900(Ω) 119894 = 1 119899 with

119903 gt 2 if 119901 le 2 119903 = 119901 if 119901 gt 2 for 119899 = 2

119903 ge 119901 119903 ge 119899 with 119903 gt 119901 if 119901 = 119899 for 119899 gt 2

(ℎ1015840

2)

119886 isin 119905(Ω) with

119905 = 119901 for 119899 = 2

119905 ge 119901 119905 ge119899

2 with 119905 gt 119901 if 119901 =

119899

2for 119899 gt 2

ess infΩ

119886 = 1198860 gt 0

(ℎ1015840

3)

In view of Theorem 1 under the assumptions (ℎ10158400)ndash(ℎ10158403) the

operator 119871 1198822119901(Ω) rarr 119871119901(Ω) is bounded

The first application is contained in Theorem 32 andCorollary 33 of [7] (see also [22] where the case 119901 = 2 isconsidered) and reads as follows

Theorem 10 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901(Ω) le 1198881003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901(Ω)

forall119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω) (51)

with 119888= 119888(Ω 119899 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) 120590119902120582

119900[(119886119894119895)119909ℎ

] 120590119903

119900[119886119894]

119905[119886]

1198860)Moreover the problem

119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω)

119871119906 = 119891 119891 isin 119871119901(Ω)

(52)

is uniquely solvable

The nodal point in achieving these results consists in theexistence of the derivatives of the 119886119894119895 Indeed this consentsto rewrite the operator 119871 in divergence form and exploit(43) in order to obtain an estimate as that in (51) but formore regular functions Then one can prove (51) by meansof an approximation argument Estimate (51) immediatelytakes to the solvability of problem (52) via a straightforwardapplication of the method of continuity along a parametersee for instance [23] and by the already known solvabilityof an opportune auxiliary problem

As second application of (43) we obtain in [8] ananalogous of Theorem 10 in a weighted framework Namelywe consider a weight function 120588119904 that is a power of a function120588 of class 1198622(Ω) such that 120588 Ω rarr R+ and

sup119909isinΩ

1003816100381610038161003816120597120572120588 (119909)

1003816100381610038161003816

120588 (119909)lt +infin forall |120572| le 2

lim|119909|rarr+infin

(120588 (119909) +1

120588 (119909)) = +infin

lim|119909|rarr+infin

120588119909(119909) + 120588

119909119909(119909)

120588 (119909)= 0

(53)

For instance one can think of 120588 as the function

120588 (119909) = (1 + |119909|2)119905

119905 isin R 0 (54)

For 119896 isin N0 119901 isin [1 +infin[ and 119904 isin R and given 120588 satisfying(53) we define the weighted Sobolev space 119882119896119901

119904(Ω) as the

space of distributions 119906 onΩ such that

119906119882119896119901

119904 (Ω)= sum

|120572|le119896

10038171003817100381710038171205881199041205971205721199061003817100381710038171003817 119871119901(Ω) lt +infin (55)

endowed with the norm in (55) Furthermore we denote theclosure of 119862infin

∘(Ω) in119882119896119901

119904(Ω) by

∘

119882119896119901

119904(Ω) and put1198820119901

119904(Ω) =

119871119901

119904(Ω)In Theorems 42 and 52 of [8] we showed the following

Theorem 11 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901

119904 (Ω)le 119888

1003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901119904 (Ω)

forall119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

(56)

Abstract and Applied Analysis 7

with 119888 = 119888(Ω 119899 119904 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) ||119886119894||119872119903(Ω) 120590119902120582119900 [(119886119894119895)119909ℎ]120590119903

119900[119886119894]

119905[119886] 1198860)

Moreover the problem

119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

119871119906 = 119891 119891 isin 119871119901

119904(Ω)

(57)

is uniquely solvable

One of the main tools in the proof of Theorem 11 is givenby the existence of a topological isomorphism from119882

119896119901

119904(Ω)

to119882119896119901(Ω) and from∘

119882119896119901

119904(Ω) to

∘

119882119896119901(Ω)This isomorphism

consents to deduce by the non-weighted bound in (51) thecorresponding weighted estimate in (56) taking into accountalso the imbedding results of Theorem 1 The existence anduniqueness of the solution of problem (57) follow then as inthe previous case from a direct application of the method ofcontinuity along a parameter by the solvability of a suitableauxiliary problem

References

[1] G Bottaro and M E Marina ldquoProblema di Dirichlet perequazioni ellittiche di tipo variazionale su insiemi non limitatirdquoBollettino dellrsquoUnione Matematica Italiana Serie 4 vol 8 pp46ndash56 1973

[2] M Transirico and M Troisi ldquoEquazioni ellittiche del secondoordine a coefficienti discontinui e di tipo variazionale in apertinon limitatirdquo Bollettino dellrsquoUnioneMatematica Italiana Serie 7vol 2 no 1 pp 385ndash398 1988

[3] M Transirico M Troisi and A Vitolo ldquoSpaces of Morreytype and elliptic equations in divergence form on unboundeddomainsrdquo Bollettino dellrsquoUnione Matematica Italiana Serie 7vol 9 no 1 pp 153ndash174 1995

[4] S Monsurro and M Transirico ldquoAn 119871119901-estimate for weak

solutions of elliptic equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 376179 15 pages 2012

[5] S Monsurro and M Transirico ldquoDirichlet problem for diver-gence form elliptic equations with discontinuous coefficientsrdquoBoundary Value Problems vol 2012 20 pages 2012

[6] S Monsurro and M Transirico ldquoA priori bounds in 119871119901 for

solutions of elliptic equations in divergence formrdquo Bulletin desSciences Mathematiques In press

[7] S Monsurro and M Transirico ldquoA 1198822119901-estimate for a class

of elliptic operatorsrdquo International Journal of Pure and AppliedMathematics vol 83 no 4 pp 489ndash499 2013

[8] S Monsurro and M Transirico ldquoA weighted 1198822119901-bound

for a class of elliptic operatorsrdquo Journal of Inequalities andApplications vol 2013 article 263 2013

[9] C Miranda ldquoSulle equazioni ellittiche del secondo ordine ditipo non variazionale a coefficienti discontinuirdquo Annali diMatematica Pura ed Applicata vol 63 no 1 pp 353ndash386 1963

[10] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines nonbornesrdquo Atti della Accademia Nazionale dei Lincei RendicontiClasse di Scienze Fisiche Matematiche e Naturali Serie VIII vol78 no 5 pp 205ndash212 1985

[11] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines

non bornes IIrdquo Atti della Accademia Nazionale dei LinceiRendiconti Classe di Scienze Fisiche Matematiche e NaturaliSerie VIII vol 79 no 6 pp 178ndash183 1985

[12] P Donato and D Giachetti ldquoQuasilinear elliptic equations withquadratic growth in unbounded domainsrdquo Nonlinear AnalysisTheoryMethods ampApplications vol 10 no 8 pp 791ndash804 1986

[13] L Caso P Cavaliere and M Transirico ldquoAn existence resultfor elliptic equations with VMO-coefficientsrdquo Journal of Mathe-matical Analysis and Applications vol 325 no 2 pp 1095ndash11022007

[14] S Boccia S Monsurro and M Transirico ldquoElliptic equationsin weighted Sobolev spaces on unbounded domainsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2008 Article ID 582435 12 pages 2008

[15] S Boccia S Monsurro and M Transirico ldquoSolvability of theDirichlet problem for elliptic equations in weighted Sobolevspaces on unbounded domainsrdquo Boundary Value Problems vol2008 Article ID 901503 13 pages 2008

[16] L Caso R DrsquoAmbrosio and S Monsurro ldquoSome remarks onspaces of Morrey typerdquo Abstract and Applied Analysis vol 2010Article ID 242079 22 pages 2010

[17] P Cavaliere M Longobardi and A Vitolo ldquoImbedding esti-mates and elliptic equations with discontinuous coefficients inunbounded domainsrdquo LeMatematiche vol 51 no 1 pp 87ndash1041996

[18] G Stampacchia ldquoLe probleme de Dirichlet pour les equationselliptiques du second ordre a coefficients discontinusrdquo Annalesde lrsquoInstitut Fourier vol 15 pp 189ndash258 1965

[19] G Stampacchia Equations Elliptiques du Second Ordre a Coef-ficients Discontinus Seminaire de Mathematiques Superieuresno 16 (Ete 1965) Les Presses de lrsquoUniversite de MontrealMontreal Canada 1966

[20] L Caso P Cavaliere and M Transirico ldquoSolvability of theDirichlet problem in1198822119901 for elliptic equations with discontin-uous coefficients in unbounded domainsrdquo Le Matematiche vol57 no 2 pp 287ndash302 2002

[21] C Miranda ldquoAlcune osservazioni sulla maggiorazione in 119871]119901

delle soluzioni deboli delle equazioni ellittiche del secondoordinerdquoAnnali diMatematica Pura ed Applicata vol 61 pp 151ndash169 1963

[22] S Monsurro M Salvato and M Transirico ldquo11988222 a prioribounds for a class of elliptic operatorsrdquo International Journalof Differential Equations vol 2011 Article ID 572824 17 pages2011

[23] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 Springer Berlin Germany2nd edition 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

Abstract and Applied Analysis 5

Using this last inequality and Holder inequality weconclude our proof since

119906119901

119871119901(Ω)le intΩ

|119906|119901minus2

(1199062

119909+ 1199062) 119889119909

le 119862119886 (119906 |119906|119901minus2

119906) = 119862intΩ

119891|119906|119901minus2

119906119889119909

le 119862intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909 le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(38)

If (ℎ1) (ℎ2) and (ℎ4) hold we consider again thefunctions 119906119904 119904 = 1 119903 obtained in correspondence ofthe solution 119906 and of 119892 as in the previous case and of 120576 asin Lemma 41 of [5] In this second case easy computationstogether with (29) give

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

intΩ

10038161003816100381610038161199061199041003816100381610038161003816119901119889119909 (39)

with 1198880 = 1198880(119903 119901)Thus from (33) we deduce that

intΩ

|119906|119901119889119909 le 1198880

119903

sum

119904=1

119862s

119903

sum

ℎ=s119886 (119906

1003816100381610038161003816119906ℎ1003816100381610038161003816119901minus2

119906ℎ)

le 1198881

119903

sum

119904=1

119886 (11990610038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904)

(40)

with 119862119904 = 119862119904(119904 119903 ] 120583) and 1198881 = 1198881(119903 119901 ] 120583)Hence by (28) and Holder inequality we obtain

119906119901

119871119901(Ω)le 1198881

119903

sum

119904=1

intΩ

11989110038161003816100381610038161199061199041003816100381610038161003816119901minus2

119906119904119889119909

le 1199031198881intΩ

10038161003816100381610038161198911003816100381610038161003816 |119906|119901minus1

119889119909

le 119903119888110038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119906

119901minus1

119871119901(Ω)

(41)

This ends the proof in view of (30)

In the later paper [6] estimate (34) has been improveddropping the hypotheses on the boundedness of 119891 and 119906 bymeans of the theorem below

Theorem 8 Assume that hypotheses (ℎ1)ndash(ℎ3) or (ℎ1) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]2 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (42)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

The proof which is different according to hypothesis (ℎ3)or (ℎ4) is essentially performed into two steps In the firststep we show some regularity results exploiting a technique

introduced by Miranda in [21] Namely we prove that if 119906 isin∘

11988212(Ω) is the solution of (22) with119891 isin 119871

2(Ω)cap119871

infin(Ω) then

the datum 119891 being more regular one also has 119906 isin 119871infin(Ω)

Thus Theorem 7 applies giving that 119906 isin 119871119901(Ω) and satisfies

(34) The second step consists in considering a datum 119891 isin

1198712(Ω) cap 119871

119901(Ω) and then one can conclude by means of some

approximation arguments see also [16]Finally in [6] we prove the main result that is the

claimed 119871119901-bound 119901 gt 1 To this aim a further assumptionon the leading coefficients is required

119886119894119895 = 119886119895119894 119894 119895 = 1 119899 (ℎ0)

Then one has the following

Theorem 9 Assume that hypotheses (ℎ0)ndash(ℎ3) or (ℎ0) (ℎ2)and (ℎ4) are satisfied If the setΩ has the uniform119862

1-regularityproperty and the datum 119891 isin 119871

2(Ω) cap 119871

119901(Ω) for some 119901 isin

]1 +infin[ then the solution 119906 of problem (22) is in 119871119901(Ω) and

119906 119871119901(Ω) le 11986210038171003817100381710038171198911003817100381710038171003817 119871119901(Ω) (43)

where119862 is a constant depending on 119899 119905 119901 ] 120583 ||119887119894minus119889119894||1198722119905120582(Ω)119894 = 1 119899

Proof For 119901 ge 2 Theorems 2 and 8 already prove the resultIt remains to show it for 1 lt 119901 lt 2

We assume that hypotheses (ℎ0)ndash(ℎ3) hold true Underhypotheses (ℎ0) (ℎ2) and (ℎ4) a similar argument withsuitable modifications can be used (we refer the reader to [6]for the details)

Let us define the bilinear form

119886lowast(119908 V) = 119886 (V 119908) 119908 V isin

∘

11988212(Ω) (44)

By (ℎ0) one has

119886lowast(119908 V)

= intΩ

(

119899

sum

119894119895=1

(119886119894119895119908119909119894+119887119895119908) V119909119895+(

119899

sum

119894=1

119889119894119908119909119894+119888119908) V)119889119909

(45)

Now consider the problem

119908 isin∘

11988212(Ω)

119886lowast(119908 V) = int

Ω

119892V 119889119909 119892 isin 1198712(Ω) cap 119871

1199011015840

(Ω)

(46)

where since 1 lt 119901 lt 2 one gets 1199011015840 = 119901(119901 minus 1) gt 2As a consequence of Theorem 2 (in the second set of

hypotheses) the solution 119908 of (46) exists and is uniqueFurthermore byTheorem 8 (in the second set of hypotheses)one also has

1199081198711199011015840

(Ω)le 119862

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω) (47)

Hence if we denote by 119906 the solution of

119906 isin∘

11988212(Ω)

119871119906 = 119891 119891 isin 1198712(Ω) cap 119871

119901(Ω)

(48)

6 Abstract and Applied Analysis

which exists and is unique in view of Theorem 2 (in the firstset of hypotheses) we obtain

intΩ

119892119906 119889119909 = 119886lowast(119908 119906) = 119886 (119906 119908) = int

Ω

119891119908 119889119909

le10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119908 119871119901

1015840

(Ω)le 119862

10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω)

(49)

Finally taking 119892 = |119906|119901minus1 sign 119906 in (49) we get the claimed

result

4 Non-Variational Problems

In this section we show two applications of ourmain estimate(43)

To this aim let 119901 gt 1 and assume that

Ω has the uniform 11986211-regularity property (ℎ

1015840

0)

Consider then the non-variational differential operator

119871 = minus

119899

sum

119894119895=1

1198861198941198951205972

120597119909119894120597119909119895+

119899

sum

119894=1

119886119894120597

120597119909119894+ 119886 (50)

with the following conditions on the leading coefficients

119886119894119895 = 119886119895119894 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(119886119894119895)119909ℎisin 119872119902120582

119900(Ω) 119894 119895 ℎ = 1 119899 with

119902 gt 2 120582 = 0 for 119899 = 2

119902 isin ] 2 119899 [ 120582 = 119899 minus 119902 for 119899 gt 2

(ℎ1015840

1)

Suppose that the lower order terms are such that

119886119894 isin 119872119903

119900(Ω) 119894 = 1 119899 with

119903 gt 2 if 119901 le 2 119903 = 119901 if 119901 gt 2 for 119899 = 2

119903 ge 119901 119903 ge 119899 with 119903 gt 119901 if 119901 = 119899 for 119899 gt 2

(ℎ1015840

2)

119886 isin 119905(Ω) with

119905 = 119901 for 119899 = 2

119905 ge 119901 119905 ge119899

2 with 119905 gt 119901 if 119901 =

119899

2for 119899 gt 2

ess infΩ

119886 = 1198860 gt 0

(ℎ1015840

3)

In view of Theorem 1 under the assumptions (ℎ10158400)ndash(ℎ10158403) the

operator 119871 1198822119901(Ω) rarr 119871119901(Ω) is bounded

The first application is contained in Theorem 32 andCorollary 33 of [7] (see also [22] where the case 119901 = 2 isconsidered) and reads as follows

Theorem 10 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901(Ω) le 1198881003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901(Ω)

forall119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω) (51)

with 119888= 119888(Ω 119899 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) 120590119902120582

119900[(119886119894119895)119909ℎ

] 120590119903

119900[119886119894]

119905[119886]

1198860)Moreover the problem

119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω)

119871119906 = 119891 119891 isin 119871119901(Ω)

(52)

is uniquely solvable

The nodal point in achieving these results consists in theexistence of the derivatives of the 119886119894119895 Indeed this consentsto rewrite the operator 119871 in divergence form and exploit(43) in order to obtain an estimate as that in (51) but formore regular functions Then one can prove (51) by meansof an approximation argument Estimate (51) immediatelytakes to the solvability of problem (52) via a straightforwardapplication of the method of continuity along a parametersee for instance [23] and by the already known solvabilityof an opportune auxiliary problem

As second application of (43) we obtain in [8] ananalogous of Theorem 10 in a weighted framework Namelywe consider a weight function 120588119904 that is a power of a function120588 of class 1198622(Ω) such that 120588 Ω rarr R+ and

sup119909isinΩ

1003816100381610038161003816120597120572120588 (119909)

1003816100381610038161003816

120588 (119909)lt +infin forall |120572| le 2

lim|119909|rarr+infin

(120588 (119909) +1

120588 (119909)) = +infin

lim|119909|rarr+infin

120588119909(119909) + 120588

119909119909(119909)

120588 (119909)= 0

(53)

For instance one can think of 120588 as the function

120588 (119909) = (1 + |119909|2)119905

119905 isin R 0 (54)

For 119896 isin N0 119901 isin [1 +infin[ and 119904 isin R and given 120588 satisfying(53) we define the weighted Sobolev space 119882119896119901

119904(Ω) as the

space of distributions 119906 onΩ such that

119906119882119896119901

119904 (Ω)= sum

|120572|le119896

10038171003817100381710038171205881199041205971205721199061003817100381710038171003817 119871119901(Ω) lt +infin (55)

endowed with the norm in (55) Furthermore we denote theclosure of 119862infin

∘(Ω) in119882119896119901

119904(Ω) by

∘

119882119896119901

119904(Ω) and put1198820119901

119904(Ω) =

119871119901

119904(Ω)In Theorems 42 and 52 of [8] we showed the following

Theorem 11 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901

119904 (Ω)le 119888

1003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901119904 (Ω)

forall119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

(56)

Abstract and Applied Analysis 7

with 119888 = 119888(Ω 119899 119904 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) ||119886119894||119872119903(Ω) 120590119902120582119900 [(119886119894119895)119909ℎ]120590119903

119900[119886119894]

119905[119886] 1198860)

Moreover the problem

119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

119871119906 = 119891 119891 isin 119871119901

119904(Ω)

(57)

is uniquely solvable

One of the main tools in the proof of Theorem 11 is givenby the existence of a topological isomorphism from119882

119896119901

119904(Ω)

to119882119896119901(Ω) and from∘

119882119896119901

119904(Ω) to

∘

119882119896119901(Ω)This isomorphism

consents to deduce by the non-weighted bound in (51) thecorresponding weighted estimate in (56) taking into accountalso the imbedding results of Theorem 1 The existence anduniqueness of the solution of problem (57) follow then as inthe previous case from a direct application of the method ofcontinuity along a parameter by the solvability of a suitableauxiliary problem

References

[1] G Bottaro and M E Marina ldquoProblema di Dirichlet perequazioni ellittiche di tipo variazionale su insiemi non limitatirdquoBollettino dellrsquoUnione Matematica Italiana Serie 4 vol 8 pp46ndash56 1973

[2] M Transirico and M Troisi ldquoEquazioni ellittiche del secondoordine a coefficienti discontinui e di tipo variazionale in apertinon limitatirdquo Bollettino dellrsquoUnioneMatematica Italiana Serie 7vol 2 no 1 pp 385ndash398 1988

[3] M Transirico M Troisi and A Vitolo ldquoSpaces of Morreytype and elliptic equations in divergence form on unboundeddomainsrdquo Bollettino dellrsquoUnione Matematica Italiana Serie 7vol 9 no 1 pp 153ndash174 1995

[4] S Monsurro and M Transirico ldquoAn 119871119901-estimate for weak

solutions of elliptic equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 376179 15 pages 2012

[5] S Monsurro and M Transirico ldquoDirichlet problem for diver-gence form elliptic equations with discontinuous coefficientsrdquoBoundary Value Problems vol 2012 20 pages 2012

[6] S Monsurro and M Transirico ldquoA priori bounds in 119871119901 for

solutions of elliptic equations in divergence formrdquo Bulletin desSciences Mathematiques In press

[7] S Monsurro and M Transirico ldquoA 1198822119901-estimate for a class

of elliptic operatorsrdquo International Journal of Pure and AppliedMathematics vol 83 no 4 pp 489ndash499 2013

[8] S Monsurro and M Transirico ldquoA weighted 1198822119901-bound

for a class of elliptic operatorsrdquo Journal of Inequalities andApplications vol 2013 article 263 2013

[9] C Miranda ldquoSulle equazioni ellittiche del secondo ordine ditipo non variazionale a coefficienti discontinuirdquo Annali diMatematica Pura ed Applicata vol 63 no 1 pp 353ndash386 1963

[10] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines nonbornesrdquo Atti della Accademia Nazionale dei Lincei RendicontiClasse di Scienze Fisiche Matematiche e Naturali Serie VIII vol78 no 5 pp 205ndash212 1985

[11] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines

non bornes IIrdquo Atti della Accademia Nazionale dei LinceiRendiconti Classe di Scienze Fisiche Matematiche e NaturaliSerie VIII vol 79 no 6 pp 178ndash183 1985

[12] P Donato and D Giachetti ldquoQuasilinear elliptic equations withquadratic growth in unbounded domainsrdquo Nonlinear AnalysisTheoryMethods ampApplications vol 10 no 8 pp 791ndash804 1986

[13] L Caso P Cavaliere and M Transirico ldquoAn existence resultfor elliptic equations with VMO-coefficientsrdquo Journal of Mathe-matical Analysis and Applications vol 325 no 2 pp 1095ndash11022007

[14] S Boccia S Monsurro and M Transirico ldquoElliptic equationsin weighted Sobolev spaces on unbounded domainsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2008 Article ID 582435 12 pages 2008

[15] S Boccia S Monsurro and M Transirico ldquoSolvability of theDirichlet problem for elliptic equations in weighted Sobolevspaces on unbounded domainsrdquo Boundary Value Problems vol2008 Article ID 901503 13 pages 2008

[16] L Caso R DrsquoAmbrosio and S Monsurro ldquoSome remarks onspaces of Morrey typerdquo Abstract and Applied Analysis vol 2010Article ID 242079 22 pages 2010

[17] P Cavaliere M Longobardi and A Vitolo ldquoImbedding esti-mates and elliptic equations with discontinuous coefficients inunbounded domainsrdquo LeMatematiche vol 51 no 1 pp 87ndash1041996

[18] G Stampacchia ldquoLe probleme de Dirichlet pour les equationselliptiques du second ordre a coefficients discontinusrdquo Annalesde lrsquoInstitut Fourier vol 15 pp 189ndash258 1965

[19] G Stampacchia Equations Elliptiques du Second Ordre a Coef-ficients Discontinus Seminaire de Mathematiques Superieuresno 16 (Ete 1965) Les Presses de lrsquoUniversite de MontrealMontreal Canada 1966

[20] L Caso P Cavaliere and M Transirico ldquoSolvability of theDirichlet problem in1198822119901 for elliptic equations with discontin-uous coefficients in unbounded domainsrdquo Le Matematiche vol57 no 2 pp 287ndash302 2002

[21] C Miranda ldquoAlcune osservazioni sulla maggiorazione in 119871]119901

delle soluzioni deboli delle equazioni ellittiche del secondoordinerdquoAnnali diMatematica Pura ed Applicata vol 61 pp 151ndash169 1963

[22] S Monsurro M Salvato and M Transirico ldquo11988222 a prioribounds for a class of elliptic operatorsrdquo International Journalof Differential Equations vol 2011 Article ID 572824 17 pages2011

[23] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 Springer Berlin Germany2nd edition 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

6 Abstract and Applied Analysis

which exists and is unique in view of Theorem 2 (in the firstset of hypotheses) we obtain

intΩ

119892119906 119889119909 = 119886lowast(119908 119906) = 119886 (119906 119908) = int

Ω

119891119908 119889119909

le10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)119908 119871119901

1015840

(Ω)le 119862

10038171003817100381710038171198911003817100381710038171003817 119871119901(Ω)

10038171003817100381710038171198921003817100381710038171003817 1198711199011015840

(Ω)

(49)

Finally taking 119892 = |119906|119901minus1 sign 119906 in (49) we get the claimed

result

4 Non-Variational Problems

In this section we show two applications of ourmain estimate(43)

To this aim let 119901 gt 1 and assume that

Ω has the uniform 11986211-regularity property (ℎ

1015840

0)

Consider then the non-variational differential operator

119871 = minus

119899

sum

119894119895=1

1198861198941198951205972

120597119909119894120597119909119895+

119899

sum

119894=1

119886119894120597

120597119909119894+ 119886 (50)

with the following conditions on the leading coefficients

119886119894119895 = 119886119895119894 isin 119871infin(Ω) 119894 119895 = 1 119899

exist] gt 0

119899

sum

119894119895=1

119886119894119895120585119894120585119895 ge ]100381610038161003816100381612058510038161003816100381610038162 ae in Ω forall120585 isin R

119899

(119886119894119895)119909ℎisin 119872119902120582

119900(Ω) 119894 119895 ℎ = 1 119899 with

119902 gt 2 120582 = 0 for 119899 = 2

119902 isin ] 2 119899 [ 120582 = 119899 minus 119902 for 119899 gt 2

(ℎ1015840

1)

Suppose that the lower order terms are such that

119886119894 isin 119872119903

119900(Ω) 119894 = 1 119899 with

119903 gt 2 if 119901 le 2 119903 = 119901 if 119901 gt 2 for 119899 = 2

119903 ge 119901 119903 ge 119899 with 119903 gt 119901 if 119901 = 119899 for 119899 gt 2

(ℎ1015840

2)

119886 isin 119905(Ω) with

119905 = 119901 for 119899 = 2

119905 ge 119901 119905 ge119899

2 with 119905 gt 119901 if 119901 =

119899

2for 119899 gt 2

ess infΩ

119886 = 1198860 gt 0

(ℎ1015840

3)

In view of Theorem 1 under the assumptions (ℎ10158400)ndash(ℎ10158403) the

operator 119871 1198822119901(Ω) rarr 119871119901(Ω) is bounded

The first application is contained in Theorem 32 andCorollary 33 of [7] (see also [22] where the case 119901 = 2 isconsidered) and reads as follows

Theorem 10 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901(Ω) le 1198881003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901(Ω)

forall119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω) (51)

with 119888= 119888(Ω 119899 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) 120590119902120582

119900[(119886119894119895)119909ℎ

] 120590119903

119900[119886119894]

119905[119886]

1198860)Moreover the problem

119906 isin 1198822119901

(Ω) cap∘

1198821119901

(Ω)

119871119906 = 119891 119891 isin 119871119901(Ω)

(52)

is uniquely solvable

The nodal point in achieving these results consists in theexistence of the derivatives of the 119886119894119895 Indeed this consentsto rewrite the operator 119871 in divergence form and exploit(43) in order to obtain an estimate as that in (51) but formore regular functions Then one can prove (51) by meansof an approximation argument Estimate (51) immediatelytakes to the solvability of problem (52) via a straightforwardapplication of the method of continuity along a parametersee for instance [23] and by the already known solvabilityof an opportune auxiliary problem

As second application of (43) we obtain in [8] ananalogous of Theorem 10 in a weighted framework Namelywe consider a weight function 120588119904 that is a power of a function120588 of class 1198622(Ω) such that 120588 Ω rarr R+ and

sup119909isinΩ

1003816100381610038161003816120597120572120588 (119909)

1003816100381610038161003816

120588 (119909)lt +infin forall |120572| le 2

lim|119909|rarr+infin

(120588 (119909) +1

120588 (119909)) = +infin

lim|119909|rarr+infin

120588119909(119909) + 120588

119909119909(119909)

120588 (119909)= 0

(53)

For instance one can think of 120588 as the function

120588 (119909) = (1 + |119909|2)119905

119905 isin R 0 (54)

For 119896 isin N0 119901 isin [1 +infin[ and 119904 isin R and given 120588 satisfying(53) we define the weighted Sobolev space 119882119896119901

119904(Ω) as the

space of distributions 119906 onΩ such that

119906119882119896119901

119904 (Ω)= sum

|120572|le119896

10038171003817100381710038171205881199041205971205721199061003817100381710038171003817 119871119901(Ω) lt +infin (55)

endowed with the norm in (55) Furthermore we denote theclosure of 119862infin

∘(Ω) in119882119896119901

119904(Ω) by

∘

119882119896119901

119904(Ω) and put1198820119901

119904(Ω) =

119871119901

119904(Ω)In Theorems 42 and 52 of [8] we showed the following

Theorem 11 Let 119871 be defined in (50) If hypotheses (ℎ10158400)ndash(ℎ10158403)

are satisfied then there exists a constant 119888 isin R+ such that

1199061198822119901

119904 (Ω)le 119888

1003817100381710038171003817100381711987111990610038171003817100381710038171003817 119871119901119904 (Ω)

forall119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

(56)

Abstract and Applied Analysis 7

with 119888 = 119888(Ω 119899 119904 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) ||119886119894||119872119903(Ω) 120590119902120582119900 [(119886119894119895)119909ℎ]120590119903

119900[119886119894]

119905[119886] 1198860)

Moreover the problem

119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

119871119906 = 119891 119891 isin 119871119901

119904(Ω)

(57)

is uniquely solvable

One of the main tools in the proof of Theorem 11 is givenby the existence of a topological isomorphism from119882

119896119901

119904(Ω)

to119882119896119901(Ω) and from∘

119882119896119901

119904(Ω) to

∘

119882119896119901(Ω)This isomorphism

consents to deduce by the non-weighted bound in (51) thecorresponding weighted estimate in (56) taking into accountalso the imbedding results of Theorem 1 The existence anduniqueness of the solution of problem (57) follow then as inthe previous case from a direct application of the method ofcontinuity along a parameter by the solvability of a suitableauxiliary problem

References

[1] G Bottaro and M E Marina ldquoProblema di Dirichlet perequazioni ellittiche di tipo variazionale su insiemi non limitatirdquoBollettino dellrsquoUnione Matematica Italiana Serie 4 vol 8 pp46ndash56 1973

[2] M Transirico and M Troisi ldquoEquazioni ellittiche del secondoordine a coefficienti discontinui e di tipo variazionale in apertinon limitatirdquo Bollettino dellrsquoUnioneMatematica Italiana Serie 7vol 2 no 1 pp 385ndash398 1988

[3] M Transirico M Troisi and A Vitolo ldquoSpaces of Morreytype and elliptic equations in divergence form on unboundeddomainsrdquo Bollettino dellrsquoUnione Matematica Italiana Serie 7vol 9 no 1 pp 153ndash174 1995

[4] S Monsurro and M Transirico ldquoAn 119871119901-estimate for weak

solutions of elliptic equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 376179 15 pages 2012

[5] S Monsurro and M Transirico ldquoDirichlet problem for diver-gence form elliptic equations with discontinuous coefficientsrdquoBoundary Value Problems vol 2012 20 pages 2012

[6] S Monsurro and M Transirico ldquoA priori bounds in 119871119901 for

solutions of elliptic equations in divergence formrdquo Bulletin desSciences Mathematiques In press

[7] S Monsurro and M Transirico ldquoA 1198822119901-estimate for a class

of elliptic operatorsrdquo International Journal of Pure and AppliedMathematics vol 83 no 4 pp 489ndash499 2013

[8] S Monsurro and M Transirico ldquoA weighted 1198822119901-bound

for a class of elliptic operatorsrdquo Journal of Inequalities andApplications vol 2013 article 263 2013

[9] C Miranda ldquoSulle equazioni ellittiche del secondo ordine ditipo non variazionale a coefficienti discontinuirdquo Annali diMatematica Pura ed Applicata vol 63 no 1 pp 353ndash386 1963

[10] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines nonbornesrdquo Atti della Accademia Nazionale dei Lincei RendicontiClasse di Scienze Fisiche Matematiche e Naturali Serie VIII vol78 no 5 pp 205ndash212 1985

[11] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines

non bornes IIrdquo Atti della Accademia Nazionale dei LinceiRendiconti Classe di Scienze Fisiche Matematiche e NaturaliSerie VIII vol 79 no 6 pp 178ndash183 1985

[12] P Donato and D Giachetti ldquoQuasilinear elliptic equations withquadratic growth in unbounded domainsrdquo Nonlinear AnalysisTheoryMethods ampApplications vol 10 no 8 pp 791ndash804 1986

[13] L Caso P Cavaliere and M Transirico ldquoAn existence resultfor elliptic equations with VMO-coefficientsrdquo Journal of Mathe-matical Analysis and Applications vol 325 no 2 pp 1095ndash11022007

[14] S Boccia S Monsurro and M Transirico ldquoElliptic equationsin weighted Sobolev spaces on unbounded domainsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2008 Article ID 582435 12 pages 2008

[15] S Boccia S Monsurro and M Transirico ldquoSolvability of theDirichlet problem for elliptic equations in weighted Sobolevspaces on unbounded domainsrdquo Boundary Value Problems vol2008 Article ID 901503 13 pages 2008

[16] L Caso R DrsquoAmbrosio and S Monsurro ldquoSome remarks onspaces of Morrey typerdquo Abstract and Applied Analysis vol 2010Article ID 242079 22 pages 2010

[17] P Cavaliere M Longobardi and A Vitolo ldquoImbedding esti-mates and elliptic equations with discontinuous coefficients inunbounded domainsrdquo LeMatematiche vol 51 no 1 pp 87ndash1041996

[18] G Stampacchia ldquoLe probleme de Dirichlet pour les equationselliptiques du second ordre a coefficients discontinusrdquo Annalesde lrsquoInstitut Fourier vol 15 pp 189ndash258 1965

[19] G Stampacchia Equations Elliptiques du Second Ordre a Coef-ficients Discontinus Seminaire de Mathematiques Superieuresno 16 (Ete 1965) Les Presses de lrsquoUniversite de MontrealMontreal Canada 1966

[20] L Caso P Cavaliere and M Transirico ldquoSolvability of theDirichlet problem in1198822119901 for elliptic equations with discontin-uous coefficients in unbounded domainsrdquo Le Matematiche vol57 no 2 pp 287ndash302 2002

[21] C Miranda ldquoAlcune osservazioni sulla maggiorazione in 119871]119901

delle soluzioni deboli delle equazioni ellittiche del secondoordinerdquoAnnali diMatematica Pura ed Applicata vol 61 pp 151ndash169 1963

[22] S Monsurro M Salvato and M Transirico ldquo11988222 a prioribounds for a class of elliptic operatorsrdquo International Journalof Differential Equations vol 2011 Article ID 572824 17 pages2011

[23] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 Springer Berlin Germany2nd edition 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

Abstract and Applied Analysis 7

with 119888 = 119888(Ω 119899 119904 ] 119901 119903 119905 ||119886119894119895||119871infin(Ω) ||119886119894||119872119903(Ω) 120590119902120582119900 [(119886119894119895)119909ℎ]120590119903

119900[119886119894]

119905[119886] 1198860)

Moreover the problem

119906 isin 1198822119901

119904(Ω) cap

∘

11988212

119904(Ω)

119871119906 = 119891 119891 isin 119871119901

119904(Ω)

(57)

is uniquely solvable

One of the main tools in the proof of Theorem 11 is givenby the existence of a topological isomorphism from119882

119896119901

119904(Ω)

to119882119896119901(Ω) and from∘

119882119896119901

119904(Ω) to

∘

119882119896119901(Ω)This isomorphism

consents to deduce by the non-weighted bound in (51) thecorresponding weighted estimate in (56) taking into accountalso the imbedding results of Theorem 1 The existence anduniqueness of the solution of problem (57) follow then as inthe previous case from a direct application of the method ofcontinuity along a parameter by the solvability of a suitableauxiliary problem

References

[1] G Bottaro and M E Marina ldquoProblema di Dirichlet perequazioni ellittiche di tipo variazionale su insiemi non limitatirdquoBollettino dellrsquoUnione Matematica Italiana Serie 4 vol 8 pp46ndash56 1973

[2] M Transirico and M Troisi ldquoEquazioni ellittiche del secondoordine a coefficienti discontinui e di tipo variazionale in apertinon limitatirdquo Bollettino dellrsquoUnioneMatematica Italiana Serie 7vol 2 no 1 pp 385ndash398 1988

[3] M Transirico M Troisi and A Vitolo ldquoSpaces of Morreytype and elliptic equations in divergence form on unboundeddomainsrdquo Bollettino dellrsquoUnione Matematica Italiana Serie 7vol 9 no 1 pp 153ndash174 1995

[4] S Monsurro and M Transirico ldquoAn 119871119901-estimate for weak

solutions of elliptic equationsrdquo Abstract and Applied Analysisvol 2012 Article ID 376179 15 pages 2012

[5] S Monsurro and M Transirico ldquoDirichlet problem for diver-gence form elliptic equations with discontinuous coefficientsrdquoBoundary Value Problems vol 2012 20 pages 2012

[6] S Monsurro and M Transirico ldquoA priori bounds in 119871119901 for

solutions of elliptic equations in divergence formrdquo Bulletin desSciences Mathematiques In press

[7] S Monsurro and M Transirico ldquoA 1198822119901-estimate for a class

of elliptic operatorsrdquo International Journal of Pure and AppliedMathematics vol 83 no 4 pp 489ndash499 2013

[8] S Monsurro and M Transirico ldquoA weighted 1198822119901-bound

for a class of elliptic operatorsrdquo Journal of Inequalities andApplications vol 2013 article 263 2013

[9] C Miranda ldquoSulle equazioni ellittiche del secondo ordine ditipo non variazionale a coefficienti discontinuirdquo Annali diMatematica Pura ed Applicata vol 63 no 1 pp 353ndash386 1963

[10] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines nonbornesrdquo Atti della Accademia Nazionale dei Lincei RendicontiClasse di Scienze Fisiche Matematiche e Naturali Serie VIII vol78 no 5 pp 205ndash212 1985

[11] P-L Lions ldquoRemarques sur les equations lineaires elliptiquesdu second ordre sous forme divergence dans les domaines

non bornes IIrdquo Atti della Accademia Nazionale dei LinceiRendiconti Classe di Scienze Fisiche Matematiche e NaturaliSerie VIII vol 79 no 6 pp 178ndash183 1985

[12] P Donato and D Giachetti ldquoQuasilinear elliptic equations withquadratic growth in unbounded domainsrdquo Nonlinear AnalysisTheoryMethods ampApplications vol 10 no 8 pp 791ndash804 1986

[13] L Caso P Cavaliere and M Transirico ldquoAn existence resultfor elliptic equations with VMO-coefficientsrdquo Journal of Mathe-matical Analysis and Applications vol 325 no 2 pp 1095ndash11022007

[14] S Boccia S Monsurro and M Transirico ldquoElliptic equationsin weighted Sobolev spaces on unbounded domainsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2008 Article ID 582435 12 pages 2008

[15] S Boccia S Monsurro and M Transirico ldquoSolvability of theDirichlet problem for elliptic equations in weighted Sobolevspaces on unbounded domainsrdquo Boundary Value Problems vol2008 Article ID 901503 13 pages 2008

[16] L Caso R DrsquoAmbrosio and S Monsurro ldquoSome remarks onspaces of Morrey typerdquo Abstract and Applied Analysis vol 2010Article ID 242079 22 pages 2010

[17] P Cavaliere M Longobardi and A Vitolo ldquoImbedding esti-mates and elliptic equations with discontinuous coefficients inunbounded domainsrdquo LeMatematiche vol 51 no 1 pp 87ndash1041996

[18] G Stampacchia ldquoLe probleme de Dirichlet pour les equationselliptiques du second ordre a coefficients discontinusrdquo Annalesde lrsquoInstitut Fourier vol 15 pp 189ndash258 1965

[19] G Stampacchia Equations Elliptiques du Second Ordre a Coef-ficients Discontinus Seminaire de Mathematiques Superieuresno 16 (Ete 1965) Les Presses de lrsquoUniversite de MontrealMontreal Canada 1966

[20] L Caso P Cavaliere and M Transirico ldquoSolvability of theDirichlet problem in1198822119901 for elliptic equations with discontin-uous coefficients in unbounded domainsrdquo Le Matematiche vol57 no 2 pp 287ndash302 2002

[21] C Miranda ldquoAlcune osservazioni sulla maggiorazione in 119871]119901

delle soluzioni deboli delle equazioni ellittiche del secondoordinerdquoAnnali diMatematica Pura ed Applicata vol 61 pp 151ndash169 1963

[22] S Monsurro M Salvato and M Transirico ldquo11988222 a prioribounds for a class of elliptic operatorsrdquo International Journalof Differential Equations vol 2011 Article ID 572824 17 pages2011

[23] D Gilbarg and N S Trudinger Elliptic Partial DifferentialEquations of Second Order vol 224 Springer Berlin Germany2nd edition 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: SaraMonsurròandMariaTransiricodownloads.hindawi.com/journals/aaa/2013/650870.pdf · weighted case, with principal coe cients having vanishing mean oscillation,has been taken into

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The vanishing corporation

The vanishing corporation

Vanishing Grace Sample

Vanishing Grace Sample

Föreläsning 11 Oscillation Kapitel 15.1-4 Oscillation harmonisk oscillation Kloss-fjädersystem

Föreläsning 11 Oscillation Kapitel 15.1-4 Oscillation harmonisk oscillation Kloss-fjädersystem

Vanishing Hitchhiker

Vanishing Hitchhiker

Vanishing Edge Pools

Vanishing Edge Pools

Vanishing art powerpoint

Vanishing art powerpoint

Vanishing points (v.m.)

Vanishing points (v.m.)

Vanishing cats.

Vanishing cats.

Vanishing TreasuresVanishing National Park ...Vanishing TreasuresVanishing National Park ServiceTreasures U.S. Department of the Interior Vanishing Treasures Program ... AZ. repairs

Vanishing TreasuresVanishing National Park ...Vanishing TreasuresVanishing National Park ServiceTreasures U.S. Department of the Interior Vanishing Treasures Program ... AZ. repairs

VANISHING TEMPLES - FPCJ

VANISHING TEMPLES - FPCJ

Vanishing Ice Catalogue

Vanishing Ice Catalogue

VANISHING ON SEVENTH STREET by Anthony Jaswinskiscreenplaysandscripts.com/script_files/V/Vanishing on 7th Street.pdf · VANISHING ON SEVENTH STREET by Anthony Jaswinski Vanishing

VANISHING ON SEVENTH STREET by Anthony Jaswinskiscreenplaysandscripts.com/script_files/V/Vanishing on 7th Street.pdf · VANISHING ON SEVENTH STREET by Anthony Jaswinski Vanishing

The Vanishing Middle

The Vanishing Middle

Will Vanishing

Will Vanishing

Vanishing soil

Vanishing soil

Vanishing quantum oscillations in Dirac semimetal ZrTe5 · oscillation phase analysis will continue to be valuable. In this work, we study the angular dependence of the quan-tum oscillations

Vanishing quantum oscillations in Dirac semimetal ZrTe5 · oscillation phase analysis will continue to be valuable. In this work, we study the angular dependence of the quan-tum oscillations

Vanishing Integrals

Vanishing Integrals

Warhol Vanishing Animals

Warhol Vanishing Animals

Vanishing Open Spaces in Florida - NumbersUSA › sites › default › files › public › assets › … · NumbersUSA Vanishing Open Spaces in Florida March 2015 vi VANISHING

Vanishing Open Spaces in Florida - NumbersUSA › sites › default › files › public › assets › … · NumbersUSA Vanishing Open Spaces in Florida March 2015 vi VANISHING

Clustering coe cients - Ohio University

Clustering coe cients - Ohio University