Existence and uniqueness of solutions for nonlinear obstacle problems with measure data

Nonlinear Analysis 43 (2001) 199–215www.elsevier.nl/locate/na

Existence and uniqueness of solutions fornonlinear obstacle problems with measure data

Chiara LeoneSISSA, Via Beirut 2-4, 34014 Trieste, Italy

Received 29 October 1998; accepted 4 November 1998

Keywords: Nonlinear obstacle problems; Measure data

1. Introduction

In this paper we consider the obstacle problem with measure data associated with anonlinear elliptic di�erential operator A of monotone type, mapping W 1;p

0 (); p¿ 1,into its dual W−1;p′

().The theory of variational inequalities (to which obstacle problems belong) has been

widely studied in the classical context of data in W−1;p′().

For any datum F ∈W−1;p′() the unilateral problem relative to A; �, and the ob-

stacle (denoted by VI(A; �; )) is the problem of �nding a function u such that

u∈W 1;p0 (); u ≥ ;

〈A(u); v− u〉 ≥ 〈F; v− u〉;∀v∈W 1;p

0 (); v ≥ :

(1.1)

This problem has a unique solution whenever there exists a function z in W 1;p0 () such

that z ≥ (by this we mean that the inequality is satis�ed Cp-quasi everywhere in ;see Section 2).Characterization 1: The solution u can be characterized (see, e.g., chapters II and

III in [10]) as the smallest function in W 1;p0 (), greater than or equal to , such that

A(u)− F = � in ;

u= 0 on @;(1.2)

for some nonnegative element � of W−1;p′().

E-mail address: [email protected] (C. Leone).

0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(99)00190 -X

200 C. Leone / Nonlinear Analysis 43 (2001) 199–215

Characterization 2: Finally, when the obstacle is Cp-quasi upper semicontinu-ous (see Section 2) u is also characterized (see, e.g., Theorem 3:2 in [1]) by thecomplementarity system

u∈W 1;p0 (); u ≥ ;

A(u) = F + �;

�∈W−1;p′(); � ≥ 0;

�({u− ¿ 0}) = 0;

(1.3)

where the pointwise values of u are de�ned Cp-quasi everywhere (see Section 2).Since � is a nonnegative element of W−1;p′

(), by the Riesz Representation Theorem,it is a nonnegative Radon measure; this explains the meaning of the last line of (1.3),which can be written also as u= �-almost everywhere in .We want to extend these results to the case where the forcing term F belongs to a

suitable subspace of the space Mb() of Radon measures � on whose total variation|�| is bounded on . As usually, we identify Mb() with the dual of the Banach spaceC0() of continuous functions that are zero on the boundary; so that the duality is〈�; u〉= ∫

u d�, for every u in C0() and the norm is ‖�‖Mb() = |�|().When �∈Mb(), the duality 〈�; u〉 has not always a meaning if u∈W 1;q

0 (); q¡N .Hence the classical formulation of the variational inequality (1.1) fails, as well as theboundary value problem (1.2) cannot be thought in the usual variational sense, evenif � = 0. Thus, we need a weaker formulation of (1.2) which allows us to overcomethis di�culty.In the linear case the duality solutions obtained by Stampacchia in [15] satisfy a

formulation which ensures uniqueness, while in the nonlinear case we may use thenotion of entropy solution introduced in [2] for L1-data and extended in [5] to boundedRadon measures vanishing on all sets of p-capacity zero. When � is a measure of thistype, the problem

A(u) = � in ;

u= 0 on @;(1.4)

has a unique entropy solution (see Section 2).In this paper the set of these measures is denoted by M

pb;0(), while by M+

b ()and by M

p;+b;0 () we denote the positive cones of Mb() and M

pb;0(), respectively.

Using the notion of entropy solution we introduce a de�nition for unilateral problemswith measure data quite similar to Characterization 1.

De�nition 1.1. We say that u is the solution of the Obstacle Problem with datum�∈M

pb;0() and obstacle (denoted by OP(A; �; )) if

1. there exists a measure �∈Mp;+b;0 () such that u is the entropy solution of (1.4)

relative to � + �, and u ≥ Cp-quasi everywhere in ,

C. Leone / Nonlinear Analysis 43 (2001) 199–215 201

2. for any �∈Mp;+b;0 () such that the entropy solution v of (1.4) relative to � + �

satis�es v ≥ Cp-quasi everywhere in , we have u ≤ v Cp-q.e. in :

By de�nition, it is clear that, if such a solution exists, it is unique.The nonnegative measure �, which is uniquely de�ned, will be called the obstacle

reaction relative to u, or the measure associated with it.By an approximation technique we will prove that for any datum � in M

pb;0() there

exists a solution in the sense of De�nition 1.1.As in the classical framework, in Section 5 we will show that the solution found

can be characterized by the complementarity system.More precisely, Theorem 2.9 shows that the solution u of OP(A; �; ) is the only

entropy solution of (1.4) relative to a measure of the form � + �; �∈Mp;+b;0 (), such

that u= ; �-almost everywhere in , and u ≥ Cp-quasi everywhere in .We also �nd a more technical characterization of the solution of OP(A; �; ), which

in the case of Cp-quasi upper bounded in turns out to be similar to (1.1). In thisframework we recover the de�nitions given by Boccardo and Gallou�et in [4] and byBoccardo and Cirmi in [3] when �=f∈L1(), and by Oppezzi and Rossi in [13,14]in a more general case.Finally, the approach to obstacle problems proposed in this paper allows us to obtain

a stability result with respect to strong convergence of data in Mb().

2. Assumptions, preliminaries, and main results

Let be a bounded, open subset of RN ; N ≥ 2. Let p and p′ be two real numbers,with p¿ 1; p′ ¿ 1 and (1=p) + (1=p′) = 1. Let a : ×RN 7→ RN be a Carath�eodoryfunction such that for almost every x in and for every �; � in RN (� 6= �):

a(x; �) · � ≥ �|�|p − h(x); (2.1)

|a(x; �)| ≤ �[k(x) + |�|p−1]; (2.2)

(a(x; �)− a(x; �)) · (�− �)¿ 0; (2.3)

a(x; 0) = 0; (2.4)

where � and � are two positive real constants, h is a nonnegative function in L1()and k is a nonnegative function in Lp′

().Thanks to hypotheses (2.1)–(2.3), the operator A : u 7→−div(a(x;∇u)) maps W 1;p

0 ()into its dual W−1;p′

(), and it is coercive, continuous, bounded, and monotone (see[12]).Let K be a compact subset of . The p-capacity of K with respect to is

de�ned as

Cp(K) = inf{∫

|∇u|p dx: u∈C∞

0 (); u ≥ �K

};


where �K is the characteristic function of K and we use the convention inf ∅ = +∞.This de�nition can be extended to any open subset B of in the following way:

Cp(B) = sup{Cp(K): K compact; K ⊆}:Finally, it is possible to de�ne the p-capacity of any set A⊆ as

Cp(A) = inf{Cp(B): B open; A⊆B}:We say that a property P(x) holds Cp-quasi everywhere (Cp-q.e.) in a set E⊆, ifit holds for all x∈E except for a subset N of E with Cp(N ) = 0.A function u : 7→ �R is said to be Cp-quasi continuous (resp. Cp-quasi upper

semicontinuous) if for every �¿ 0 there exists a set E⊆, with Cp(E)¡�, such thatthe restriction of u to \ E is a continuous (resp. upper semicontinuous) functionwith values in �R. It is well known that every u∈W 1;p

0 () has a Cp-quasi continuousrepresentative, which is uniquely de�ned (and �nite) up to a set of p-capacity zero. Inthe sequel we shall always identify u with its Cp-quasi continuous representative, sothat the pointwise values of a function u∈W 1;p

0 () are de�ned Cp-quasi everywhere.A set E⊆ is said to be Cp-quasi open if for every �¿ 0 there exists an open set

U such that E⊆U ⊆ and Cp(U \ E) ≤ �.Let Mp

b;0() be the set of measures in Mb() which are “absolutely continuous”with respect to the p-capacity, that is a measure �∈Mb() belongs to M

pb;0() if and

only if �(A) = 0 for every Borel set A⊆ such that Cp(A) = 0.It is well known that, if � belongs to W−1;p′

() ∩Mb(), then � is in Mpb;0(),

every u in W 1;p0 () ∩ L∞() is summable with respect to � and

〈�; u〉=∫u d�;

where 〈·; ·〉 denotes the duality pairing between W−1;p′() and W 1;p

0 (), while in theright-hand side u denotes the Cp-quasi continuous representative and, consequently, thepointwise values of u are de�ned �-almost everywhere.A decomposition theorem for measures in M

pb;0() is known (see Theorem 2:1

in [5]): every measure that is zero on the sets of p-capacity zero can be split intothe sum of an element in W−1;p′

() and of a function in L1(), and, conversely,every measure in L1() +W−1;p′

() is zero on the sets of p-capacity zero. So, if ameasure � belongs to M

pb;0(), then every u∈W 1;p

0 () ∩ L∞() is summable withrespect to �.For every k ¿ 0 we de�ne the truncation function Tk : R 7→ R by

Tk(t) =

{t if |t| ≤ k;

k sign(t) if |t|¿k:

Let us consider the space T1;p0 () of all functions u : 7→ �R which are almost

everywhere �nite and such that Tk(u)∈W 1;p0 () for every k ¿ 0. It is easy to see that

every function u∈T1;p0 () has a Cp-quasi continuous representative with values in �R,

that will always be identi�ed with the function u. Moreover, for every u∈T1;p0 ()


there exists a measurable function � : 7→ RN such that ∇Tk(u) = ��{|u|≤k} a.e. in (see Lemma 2:1 in [2]). This function �, which is unique up to almost everywhereequivalence, will be denoted by ∇u. Note that ∇u coincides with the distributionalgradient of u whenever u∈T

1;p0 () ∩ L1loc() and ∇u∈L1loc(;RN ).

We are now in position to recall the notion of entropy solution introduced in [2]for L1 data and extended to measures in M

pb;0() in [5], which ensures us that, when

�∈Mpb;0(), the equation

A(u) = � in

u= 0 on @;(2.5)

has a unique entropy solution in the sense of the following de�nition.

De�nition 2.1. Let �∈Mpb;0(). A function u is an entropy solution of problem (2.5)

if u belongs to T1;p0 (), and∫

a(x;∇u)∇Tk(u− ’) dx ≤

∫Tk(u− ’) d� (2.6)

for every ’ in W 1;p0 () ∩ L∞() and every k ¿ 0.

Remark 2.2. Actually, it is possible to prove that equality holds in (2.6) (see [11]).

Observe that, if u is the entropy solution of (2.5), then (the Cp-quasi continuousrepresentative of) u is �nite up to a set of p-capacity zero.We recall also the following stability result (see Theorem 1:2 in [11]):

Theorem 2.3. Let {fn}⊂L1() and {Fn}⊂W−1;p′() be such that

fn * f weakly in L1();

Fn → F strongly in W−1;p′();

let un be the entropy solutions of (2:5) relative to � = fn + Fn; and let u be theentropy solution of (2:5) relative to � = f + F . Then

limn→∞ Tk(un) = Tk(u) strongly in W 1;p

0 ();

for every k ¿ 0.

Remark 2.4. Actually, in [2,5,11] the assumptions on the operator A are slightly di�er-ent. Indeed, in these papers, the operator A satis�es (2.1) with h=0 (this fact impliescondition (2.4)). On the other hand, nothing essential changes in the proofs of the ex-istence, uniqueness and stability results, if we assume (2.1) with a general h∈L1(),and we add hypothesis (2.4).Finally, we consider a function : 7→ �R such that

≤ u� Cp-q:e: in ; (2.7)


where � is an element of W−1;p′() ∩Mb() and u� is the variational solution of

A(u) = � in ;

u= 0 on @:

Moreover, we de�ne the set

K () := {z ∈T1;p0 (): z ≥ Cp-q:e: in }:

Without loss of generality, we may suppose that is Cp-quasi upper semicontinuousthanks to the following Proposition (see Proposition 1:5 in [8]).

Proposition 2.5. Let : 7→ �R satisfying∃ z ∈W 1;p

0 () such that z ≥ Cp-q:e: in : (2.8)

Then there exists a Cp-quasi upper semicontinuous function ̂ : 7→ �R such that

1. ̂ ≥ Cp-q.e. in ;2. if ’ : 7→ �R is Cp-quasi upper semicontinuous and ’ ≥ Cp-q.e. in ; then

’ ≥ ̂ Cp-q.e. in .

Thus, in particular, K () = K ̂ ().Observe that assumption (2.7) is satis�ed if (2.8) holds, and there exists a compact

J ⊂ such that ≤ 0 in \ J . Indeed, we take as � the obstacle reaction correspond-ing to the solution of VI(A; 0; ). Then, by (1.3) supp�⊂ J , and �∈W−1;p′

() ∩Mb(). On the other hand, Example 5:3 in [6] shows that, in general, (2.8) does notimply (2.7).In De�nition 1.1 we speci�ed the formulation of obstacle problems we will adopt

in this paper. Let us note, however, that in this de�nition, since u and v are Cp-quasicontinuous, it is enough to prove u ≤ v a.e. in to obtain also the inequality Cp-q.e.in .In Section 4 we will prove the following existence theorem.

Theorem 2.6. Let satisfy (2:7) and let �∈Mpb;0(). Then there exists a unique

solution of OP(A; �; ). Moreover the corresponding obstacle reaction � satis�es

‖�‖Mb() ≤ ‖(� − �)−‖Mb(): (2.9)

In Section 5 we will prove also the following theorem about continuous dependenceon data.

Theorem 2.7. Let satisfy hypothesis (2:7); let �n; �∈Mpb;0() and �n and � the

measures associated to the solution un and u of OP(A; �n; ) and OP(A; �; ); respec-tively. If �n → � strongly in Mb(); then �n → � strongly in Mb(). Moreover; forevery k ¿ 0; Tk(un) converges to Tk(u) strongly in W 1;p

0 ().

Remark 2.8. Under slightly stronger hypotheses on the obstacle, when the data areL1() functions, the solutions considered in this paper coincide, for uniqueness reasons,


with those given by Boccardo and Gallou�et in [4] and by Boccardo and Cirmi in[3]. Indeed, these solutions are obtained as limit of solutions of variational obstacleproblems, whose data are smooth and converge strongly in Mb(). By Theorem 2.7these solution satisfy De�nition 1.1.Finally, we will show that the solution found can be characterized by the comple-

mentarity system.

Theorem 2.9. Let � be in Mpb;0() and satisfy (2:7); then the following statements

are equivalent:(1) u is the solution of OP(A; �; ) and � is the associated obstacle reaction;(2) u∈K (); �∈M

p;+b;0 (); u is the entropy solution of (2:5) relative to �+ �; and∫

Tk(u− ’) d� ≤

∫Tk(v− ’) d�;

∀’∈W 1;p0 () ∩ L∞(); ∀v∈K ();

(2.10)

(3) u∈K (); �∈Mp;+b;0 (); u is the entropy solution of (2:5) relative to �+ �; and

u= �-a:e: in : (2.11)

Remark 2.10. Observe that if is Cp-q.e. upper bounded, we can consider in (2.10)’∈W 1;p

0 ()∩L∞(); ’ ≥ Cp-q.e. in and v=’, so that, taking into account thatu is the entropy solution of (2.5) relative to � + �; u satis�es∫

a(x;∇u)∇Tk(u− ’) dx ≤

∫Tk(u− ’) d�; (2.12)

which is quite similar to the usual variational formulation (1.1). Formula (2.12) wasjust obtained in [3] when the datum � is a function in L1(). This is an alternativeproof of the fact that a solution of OP(A; �; ) coincides with that given by Boccardoand Gallou�et in [4], and by Boccardo and Cirmi in [3].

At the end of Section 5 we will show that a solution of OP(A; �; ) is also arenormalized solution of the obstacle problem, according to the de�nition of Oppezziand Rossi [13,14].

3. Preparatory results

In Section 2 we mentioned the decomposition result for Radon measures that do notcharge the sets of p-capacity zero (see Theorem 2:1 in [5]). Starting from this we canobtain the following theorem.

Theorem 3.1. Let �∈Mpb;0(); then for every �¿ 0 there exist f� ∈L1() and

F� ∈W−1;p′() ∩Mb() such that � = f� + F� and

‖f�‖L1() ≤ ‖�‖Mb(); ‖F�‖W−1;p′ () ≤ �:


Proof. The proof of Theorem 2:1 in [5] shows that if � is an element of Mpb;0(),

then � can be written as � = f + F , where f∈L1(); F ∈W−1;p′(), and

‖f‖L1() ≤ ‖�‖Mb(); ‖F‖W−1;p′ () ≤ 1:If we write � as � (�=�) and we apply the same result to �=�, we easily conclude.

Corollary 3.2. Let �n; �∈Mpb;0() be such that �n converges to � in the strong

topology of Mb(). Then there exist fn; f∈L1() and Fn; F ∈W−1;p′()∩Mb()

such that

fn → f strongly in L1();

Fn → F strongly in W−1;p′();

where �n = fn + Fn and � = f + F .

Proof. Thanks to the Theorem 3.1 the sequence �n−� can be decomposed as �n−�=gn + Gn, where

‖gn‖L1() ≤ ‖�n − �‖Mb(); ‖Gn‖W−1;p′ () ≤ �n;

with �n tending to zero as n tends to in�nity. Hence, if � = f + F , where f∈L1()and F ∈W−1;p′

(), we can de�ne fn :=f + gn and Fn :=F + Gn, and we con-clude.

Remark 3.3. By the previous corollary we obtain that the strong convergence inMb()is su�cient to apply Theorem 2.3 about continuous dependence of entropy solutions.

As concerning the approximation of measures in Mpb;0(), it is useful to state a

lemma, which is quite simple, but is proved here for the sake of completeness.

Lemma 3.4. Let �i ∈Mpb;0() (i = 1; 2) be such that �1 ≤ �2; then there exists

�ni ∈W−1;p′

() ∩ Mb() (i = 1; 2) such that �ni converges to �i strongly in Mb()

and �n1 ≤ �n

2.

Proof. It su�ces to consider the decomposition of �2 − �1 in the sense of Theorem2:2 in [7]:

0 ≤ �2 − �1 = h ;

where ∈W−1;p′() ∩ M+

b () and h∈L1(; ); h ≥ 0. On the other hand, �1 de-composes as �1 = h1 1, with 1 ∈W−1;p′

() ∩ M+b () and h1 ∈L1(; ). We take

�n1 :=Tn(h1) 1 and �n

2 :=Tn(h) + �n1, and we can conclude.

Remark 3.5. The proof of the previous lemma shows that, if �∈Mp;+b;0 (), then there

exists a nondecreasing sequence �n ∈W−1;p′() ∩ M+

b () such that �n converges to� strongly in Mb().


Now we state a comparison principle concerning the entropy solutions.

Theorem 3.6. For i=1; 2; let ui be the entropy solution of (2:5) relative to �i ∈Mpb;0().

Suppose that �1 ≤ �2; then u1 ≤ u2 almost everywhere in .

Proof. The result is well known when �i ∈W−1;p′()∩Mb() (using the monotonicity

of A). Using Lemma 3.4 together with Theorem 2.3 we can conclude.

Now we give a result concerning the solutions of obstacle problems in the variationalframework.

Theorem 3.7. Let satisfy hypothesis (2:7) and let � in W−1;p′() ∩ Mb(). Let

u be the solution of VI(A; �; ) and � be the obstacle reaction relative to u. Then �satis�es (2:9).

Proof. By an approximation argument we may suppose that �+ and �− (the positiveand the negative part of �, respectively) belong to W−1;p′

() (as in Theorem 2:2 ofDall’Aglio and Leone [6]).Moreover we observe that we can consider, without loss of generality, the case of a

nonpositive obstacle. Suppose that we have proved inequality (2.9) in this case (with� = 0). Now, if is a general obstacle satisfying hypothesis (2.7), we consider thenew obstacle − u�, which is nonpositive, and the operator B(v) = −div(a(x;∇v +∇u�) − a(x;∇u�)), which is of the same type of A. The measure � associated withthe solution v of VI(B; � − �; − u�) satis�es the inequality (2.9). Now it is easy tocheck that the function u= v + u� is the solution of VI(A; �; ) and � is the measureassociated with u.The proof of (2.9) for a nonpositive obstacle (�= 0) consists of two steps.Step 1. Suppose that there exists a positive number � such that ≤ −� Cp-q.e.

in .Observe that the function u�+ , solution of

A(u�+) = �+ in

u�+ = 0 on @;

is nonnegative and hence greater than or equal to , belongs to W 1;p0 (), and

A(u�+)− � = �− in :

So u ≤ u�+ (by Characterization 1). Let us consider the truncated function T�(u�+−u)= (u�+ − u) ∧ �; it is nonnegative and less than or equal to �.Let us compute 〈�; T�(u�+ − u)〉 in the duality between W−1;p′

() and W 1;p0 ():

〈�; T�(u�+ − u)〉= 〈A(u); T�(u�+ − u)〉 − 〈�; T�(u�+ − u)〉= 〈A(u)− A(u�+); T�(u�+ − u)〉+ 〈�−; T�(u�+ − u)〉≤ 〈�−; T�(u�+−u)〉;


where in the last inequality we used the monotonicity of A. Then, using the Cp-quasi-continuous representatives of functions in W 1;p

0 (), we can write∫T�(u�+ − u) d� ≤

∫T�(u�+ − u) d�−:

By the positivity of the measures � and �− and the properties of the truncated function,we deduce that

�({u�+ − u¿�}) ≤ �−():

When � tends to 0, the previous formula becomes

�({u�+ − u¿ 0}) ≤ �−():

Now, if we prove that �({u�+ = u}) = 0 we get the result. It su�ces to observe that�({u�+ = u}) ≤ �({u ≥ 0}) ≤ �({u¿ }), which is zero thanks to (1.3).Step 2: Suppose only that ≤ 0 Cp-q.e. in .Let us de�ne, for all �¿ 0, the sequence of functions � := − �, and consider

the solutions u� of VI(A; �; �). If we call �� the measures associated with u�, by theprevious step we get ��() ≤ �−().As � tends to zero the solution u� tends to u strongly in W 1;p

0 (), hence �� tendsto � strongly in W−1;p′

(), and this implies �() ≤ �−().

We compare now the two problems VI(A; �; ) and OP(A; �; ) when the forcingterm � is an element of W−1;p′

() ∩Mb().

Proposition 3.8. Let � be an element of W−1;p′()∩Mb() and let satisfy (2:7);

then the solution of VI(A; �; ) is the solution of OP(A; �; ).

Proof. Let u be the solution of VI(A; �; ) and � be the corresponding obstacle reaction,which is a nonnegative element of W−1;p′

(). Owing to Theorem 3.7 it belongs toMb(), hence to M

p;+b;0 (). Thus u is the entropy solution of (2.5) relative to � + �.

Take v∈K () the entropy solution of (2.5) corresponding to �+ �, where � belongsto M

p;+b;0 (). We want to prove that u ≤ v almost everywhere in . Consider �k an

approximation of � (see Lemma 3.4 and Remark 3.5):

�k ∈W−1;p′() ∩Mb(); �k ≥ 0; �k ↗ � strongly in Mb()

and vk the solution of (2.5) corresponding to � + �k . Thanks to Theorem 3.6 the se-quence vk is nondecreasing and tends to v in the sense of Theorem 2.3 (see Remark3.3). Thus, in particular, vk tends to v; Cp-q.e. in . Let k= ∧vk and denote the solu-tions of VI(A; �; k) by uk . Naturally, from the minimality of uk (see Characterization1), we deduce

uk ≤ vk a:e: in :

Since uk ≤ uk+1 Cp-q.e. in ; uk converges to a function u∗ Cp-q.e. in . Thus, u∗ ≥ Cp-q.e. in . It is easy to check that uk is bounded in W 1;p

0 (); thanks to Lemma1:2 in [9], u∗ is (the Cp-quasi continuous representative of) a function of W 1;p

0 ()


and uk converges to u∗ weakly in W 1;p0 (). Moreover, it can be easily proved that

u∗ satis�es the variational formulation (1.1) and, consequently, u∗ coincides with u.Hence, passing to the limit, we conclude that u ≤ v a.e. in .

Remaining in the variational framework we state a result that generalizes Lemma7:3 in [6] to the nonlinear operator A. We omit the proof, because we may use thesame tools as in the linear case. In Section 4 we will extend it to general measuresin M

pb;0().

Lemma 3.9. Let satisfy (2:7) and let �1; �2 ∈W−1;p′()∩Mb(). Let �1 and �2 be

the reactions of the obstacle corresponding to the solutions u1 and u2 of VI(A; �1; )and VI(A; �2; ); respectively. If �1 ≤ �2; then �1 ≥ �2.

4. Proof of the existence theorem

In this section we will prove Theorem 2.6; to simplify the exposition, it is convenientto divide the proof into various lemma.

Lemma 4.1. Let satisfy (2:7) and let {�n}⊂Mpb;0() be a nondecreasing sequence

of measures converging to � strongly in Mb(); suppose that; for every n; problemOP(A; �n; ) has a solution un and let �n be the measure associated with it. If �n

converges to � strongly in Mb(); then there exists a solution u of OP(A; �; ); and� is the obstacle reaction relative to u.

Proof. First we observe that � and � belong to Mpb;0(), by the strong convergence of

�n and �n in Mb(). We can use Theorem 2.3 about continuous dependence of entropysolutions (see Remark 3.3). For every k ¿ 0, the sequence Tk(un) converges to Tk(u) inthe strong topology of W 1;p

0 (), u being the entropy solution of (2.5) relative to �+�.The strong convergence in W 1;p

0 () implies that, up to a subsequence, still denoted byun; Tk(un) converges to Tk(u); Cp-q.e. in ; thus Tk(u) ≥ Tk( ); Cp-q.e. in , and,letting k tend to in�nity, u ≥ ; Cp-q.e. in . Let us take now the entropy solutionv of (2.5) relative to � + �, with � in M

p;+b;0 (), and assume that v ≥ ; Cp-q.e.

in . Since � ≥ �n we can write � + � = �n + �n, with �n in Mp;+b;0 (). As un is

the solution of OP(A; �n; ), we obtain that un ≤ v a.e. in , and, passing to thelimit, u ≤ v a.e. in . In conclusion, u is the solution of OP(A; �; ), according toDe�nition 1.1.

Lemma 4.2. Let satisfy (2:7) and let {�n}⊂Mpb;0() be a nonincreasing sequence

of measures converging to � strongly in Mb(); suppose that; for every n; problemOP(A; �n; ) has a solution un and let �n be the measure associated with it. If �n

converges to � strongly in Mb(); then there exists a solution u of OP(A; �; ); and� is the obstacle reaction relative to u.


Proof. As in the previous lemma, for every k ¿ 0; Tk(un) converges to Tk(u) in thestrong topology of W 1;p

0 (); u being the entropy solution of (2.5) relative to � + �;moreover u ≥ ; Cp-q.e. in . Let us take the entropy solution v of (2.5) relative to�+ �, with � in M

p;+b;0 (), and assume that v ≥ ; Cp-q.e. in . De�ne the sequence

vn as the entropy solution of (2.5) relative to �n+ �. Since �n ≥ �, by the comparisonprinciple of entropy solutions (see Theorem 3.6), we obtain vn ≥ v ≥ , and, thanks toTheorem 2.3 and Remark 3.3, vn converges to v a.e in . By the de�nition of ObstacleProblems (De�nition 1.1) we have that un ≤ vn a.e. in , and, passing to the limit,u ≤ v a.e. in . Thus u is the solution of OP(A; �; ).

Lemma 4.3. Let satisfy (2:7) and let �∈Mpb;0() with �− ∈W−1;p′

() ∩Mb().Then there exists a solution of OP(A; �; ); and the corresponding obstacle reactionsatis�es (2:9).

Proof. Since �+ is a nonnegative measure in Mpb;0(), thanks to Remark 3.5, there ex-

ists a nondecreasing sequence �+n ∈W−1;p′()∩M+

b (), converging to �+ in the strongtopology of Mb(). Let us de�ne �n := �+n −�−, so that �n ∈W−1;p′

()∩Mb() and�n is a nondecreasing sequence converging to � strongly in Mb(). By Proposition 3.8the solution un of VI(A; �n; ) is a solution of OP(A; �n; ). Thanks to Theorem 3.7the corresponding obstacle reaction �n satis�es

‖�n‖Mb() ≤ ‖(�n − �)−‖Mb(): (4.1)

As �n ≤ �n+1, by Lemma 3.9 �n ≥ �n+1. Hence, if we de�ne

�(B) := limn→∞ �n(B) for every B Borel set in ;

we know from measure theory that � is a nonnegative Borel measure, it is bounded be-cause �n is nonincreasing, and it is in M

pb;0(), since all �n are. Besides, �n converges

to � strongly in Mb(). By Lemma 4.1, there exists u solution of OP(A; �; ), and� is the measure associated with it. Moreover, passing to the limit in (4.1), weobtain (2.9).

Lemma 4.4. Let satisfy (2:7) and let �1; �2 ∈Mpb;0(); with �−

1 ; �−2 ∈W−1;p′

()∩Mb(). Let �1 and �2 be the reactions of the obstacles corresponding to the solutionsu1 and u2 of OP(A; �1; ) and OP(A; �2; ); respectively. If �1 ≤ �2; then �1 ≥ �2.

Proof. Since �+1 and �2− �1 are nonnegative measures in Mpb;0(), owing to Remark

3.5 there exist two nondecreasing sequences �+1; n; �n ∈W−1; p′()∩M+

b (), convergingstrongly in Mb() to �+1 and �2−�1, respectively. Let us de�ne �1; n :=�+1; n−�−

1 and�2; n :=�n + �1; n; thus, for i = 1; 2, �i; n belongs to W−1;p′

() ∩ Mb(), and �1; n isa nondecreasing sequence converging to �1 strongly in Mb(). For i = 1; 2, considerthe solution ui; n of VI(A; �i; n; ) and the corresponding obstacle reaction �i; n. Since�1; n ≤ �2; n, by Lemma 3.9 �1; n ≥ �2; n. Using the same arguments of the proof ofLemma 4.3, we obtain that, for i= 1; 2; �i; n converges to �i strongly in Mb(). Thuswe get the result.


Proof of Theorem 2.6. Consider �− and approximate it in the strong topology ofMb() with a nondecreasing sequence �−

n ∈W−1;p′() ∩ M+

b () (see Remark 3.5).De�ning �n :=�+−�−

n , by Lemma 4.3 we can consider the solutions un of OP(A; �n; )and the measures �n associated with them. By Lemma 4.3 we know that �n satis�es

‖�n‖Mb() ≤ ‖(�n − �)−‖Mb(): (4.2)

Since �n is nonincreasing �n is nondecreasing, by Lemma 4.4. Thus, de�ning

�(B) := limn→∞ �n(B) for every B Borel set in ;

we know from measure theory that � is a nonnegative Borel measure, it is boundedbecause �n satis�es (4.2), and it is in M

pb;0(), since all �n are. Moreover, �n converges

to � strongly in Mb(). By Lemma 4.2, there exists u solution of OP(A; �; ), and� is the corresponding obstacle reaction. Moreover, passing to the limit in (4.2) weobtain (2.9).

Corollary 4.5. Let satisfy (2:7) and let �1; �2 ∈Mpb;0(). Let �1 and �2 be the

reactions of the obstacle corresponding to the solutions u1 and u2 of OP(A; �1; )and OP(A; �2; ); respectively. If �1 ≤ �2 then �1 ≥ �2.

Proof. It su�ces to consider �−1 and approximate it in the strong topology of Mb()

with a nondecreasing sequence �−1; n ∈W−1;p′

()∩M+b () (see Remark 3.5). De�ning

�1; n :=�+1 −�−1; n and �2; n :=�1; n+�2−�1, for i=1; 2 we have that �−

i; n ∈W−1;p′()∩

M+b (), and �i; n is a nonincreasing sequence converging to �i strongly in Mb().

Moreover, �1; n ≤ �2; n. We consider the solutions ui; n of OP(A; �i; n; ) and the corre-sponding obstacle reactions �i; n, which are nondecreasing in n and satisfy �1; n ≥ �2; n(by Lemma 4.4). Using the same tools of the proof of Theorem 2.6, we obtain that�i; n converges to �i strongly in Mb(). In conclusion �1 ≥ �2.

5. Proof of the stability result and of the complementarity conditions

In this section we will prove Theorems 2.7 and 2.9.

Proof of Theorem 2.7. Since �n converges to � strongly in Mb(), there exists a sub-sequence �nj of �n such that

+∞∑j=1

‖�nj − �‖Mb()¡+∞:

Let us de�ne �′j :=� −∑+∞

i=j (�ni − �)− and �′′j :=� +

∑+∞i=j (�ni − �)+; it is easy to

check that

�′j ↗ �; �′′

j ↘ � strongly in Mb()

and

�′j ≤ �nj ≤ �′′

j :


For every j≥ 1, let u′j and u′′j be the solutions of OP(A; �′j; ) and OP(A; �′′

j ; ),respectively, and let �′j and �′′j be the corresponding associated measures. Reasoningas in the proof of Theorem 2.6, we obtain that u′j and u′′j converge to u (in the senseof Theorem 2.3), while �′j and �′′j converge to � in the strong topology of Mb(). Onthe other hand, thanks to Corollary 4.5, we have �′′j ≤ �nj ≤ �′j, so that �nj convergesto � strongly in Mb(). Finally, since the result does not depend on the subsequenceall �n converges to � strongly in Mb().By Remark 3.3 we can apply Theorem 2.3 to obtain the convergence of Tk(un) to

Tk(u) in the strong topology of W1;p0 (), for every k ¿ 0.

Proof of Theorem 2.9. We will divide the proof in three steps.Step 1: (1) ⇒ (2). It su�ces to prove (2.10). We proceed by an approximation

argument. Let �n ∈W−1;p′() ∩Mb() be such that �n converges to � in the strong

topology of Mb() (see Lemma 3.4). If we consider the solution un of VI(A; �n; )and the measure �n associated to it, thanks to (1.1) we have

〈�n; w − un〉 ≥ 0;∀w∈K () ∩W 1;p

0 (): (5.1)

We choose as test function in (5.1) w=un−Tk(un−’)+Tk(v−’), where ’∈W 1;p0 ()∩

L∞() and v∈K (), and we obtain

〈�n; Tk(un − ’)〉 ≤ 〈�n; Tk(v− ’)〉:Recalling that �∈M

pb;0(), thanks to Theorem 2.7 we can pass to the limit and we

get the result.Step 2: (2) ⇒ (3). Let t be a positive real number. Observe that the set {u −

¿ t} is Cp-quasi open, because u is Cp-quasi continuous and is Cp-quasi uppersemicontinuous. Thanks to Lemma 1:5 in [7] there exists an increasing sequence vnof nonnegative functions in W 1;p

0 () which converges to �{u− ¿t} Cp-q.e. in . Thefunction u− tvn belongs to K (), thus we can apply (2.10) with v=u− tvn; observingthat u− tvn ≤ u and � is a nonnegative measure, we get∫

Tk(u− ’) d�=

∫Tk(u− tvn − ’) d�

for every ’ in W 1;p0 () ∩ L∞().

Passing to the limit as n goes to in�nity, we get∫{u− ¿t}

Tk(u− ’) d�=∫{u− ¿t}

Tk(u− t − ’) d�: (5.2)

Now we choose ’= Th(u) in (5.2). Let us estimate the left-hand side of (5.2):∣∣∣∣∣∫{u− ¿t}

Tk(u− Th(u)) d�

∣∣∣∣∣ ≤ k�({|u|¿h});


which tends to zero if h tends to in�nity (recall that u is �nite up to a set of p-capacityzero and � vanishes on the sets of p-capacity zero).In the same way we can split the integral in the right-hand side into two parts:

∫{u− ¿t}∩{|u|≤h}

Tk(−t) d�+∫{u− ¿t}∩{|u|¿h}

Tk(u− t − Th(u)) d�

as before, the second integral tends to zero if h goes to in�nity.In conclusion, we obtain Tk(t) �({u− ¿ t}) = 0, from which �({u− ¿ t}) = 0,

for every t ¿ 0. Letting t tend to zero, we get the result.Step 3: (3)⇒ (1). We have to prove that, for any � inMp;+

b;0 () such that the entropysolution v relative to �+ � belongs to K (), we have u ≤ v almost everywhere in .The procedure used in [2] allows to obtain that u satis�es (in particular)

∫a(x;∇u)∇Tk(u− ’)+ dx −

∫Tk(u− ’)+ d� =

∫Tk(u− ’)+ d�; (5.3)

for every ’ in W 1;p0 () ∩ L∞().

Similarly, v satis�es

−∫a(x;∇v)∇Tk(v− ’)− dx +

∫Tk(v− ’)− d� =−

∫Tk(v− ’)− d�: (5.4)

We choose ’= Th(v) in (5.3) and ’= Th(u) in (5.4), and we add the two equations.For the left-hand side we can use the same tools of the proof of uniqueness of

entropy solutions (see Theorem 3:3 in [5]); thus we obtain

∫{0¡u−v≤k; |u|≤h; |v|≤h}

(a(x;∇u)− a(x;∇v))∇(u− v) dx

≤ !k(h) +∫Tk(u− Th(v))+ d�−

∫Tk(v− Th(u))− d�; (5.5)

where !k(h) tends to zero if h goes to in�nity.Since � is a nonnegative measure we can rewrite (5.5) as

∫{0¡u−v≤k; |u|≤h; |v|≤h}

(a(x;∇u)− a(x;∇v))∇(u− v) dx

≤ !k(h) +∫Tk(u− Th(v))+ d�;

now we let h tend to in�nity, so that

∫{0¡u−v≤k}

(a(x;∇u)− a(x;∇v))∇(u− v) dx ≤∫Tk(u− v)+ d�:


Observing that {u− v¿ 0}⊆{u− ¿ 0}, since v∈K (), the term in the right-handside is zero, by hypothesis (2.11). In conclusion

∫{0¡u−v≤k}

(a(x;∇u)− a(x;∇v))∇(u− v) dx ≤ 0;

for every k ¿ 0; by (2.3) this implies u ≤ v almost everywhere in .

Remark 5.1. If is Cp-q.e. upper bounded, we point out that the solution u ofOP(A; �; ) satis�es

∫a(x;∇u)∇(h(u)(u− ’)) dx ≤

∫h(u)(u− ’) d� (5.6)

for every ’∈W 1;p0 () ∩ L∞(), ’ ≥ Cp-q.e. in , and for every h∈C1c (R), h ≥ 0

in R. Indeed, working as in the proof of the �rst step of Theorem 2.9, we choose in(5.1) w=un−�h(un)(un−’), with h and ’ as before and � a positive constant such that�‖h‖∞ ≤ 1. Passing to the limit we obtain (5.6). Thus u is also a renormalized solutionof the obstacle problem according to the de�nition of Oppezzi and Rossi [13,14].

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Existence and uniqueness of solutions for nonlinear obstacle problems with measure data

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