Một số vấn đề về không gian Sobolev

7/28/2019 Mt s vn v khng gian Sobolev

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Li cm t

Thi gian thm thot thoi a, chp mt m em hon

thnh bn nm i hc. Nh ngy no, u kha hc, ba cn

a n trng gp thy c mi, bn b mi vi bao bng lo

lng. Vy m cui cng em cng tri qua bn nm hc. Bn

nm hc tp vi bit bao kh khn, vt v, c nhng lc vp

ng em tng nh mnh khng th vt qua. Nhng mong

mun c lm lun vn khi tt nghip thc y em phn

u nhiu trong hc tp. Cui cng vi kt qu t c trong

cc nm u, em c b mn phn cng lm lun vn di s

hng dn ca thy Phm Gia Khnh. c lm lun vn l

mt nim vui, nim vinh d ln i vi em. Nhng bn cnh

cng c khng t ni lo v gp nhiu kh khn, no l khan

him ti liu, thi gian hn hp, m kin thc th mi v tng

i kh Nhng vi kin thc m em c thy c b mn

trang b trong cc nm qua cng vi s hng dn nhit tnh

ca thy Phm Gia Khnh cng nh s ng vin gip ca

gia nh, bn b, cui cng cun lun vn cng c hon

thnh. Em xin gi li cm n chn thnh nht n thy hng

dn, cc thy c khc trong b mn, cng gia nh v bn b.

Cn Th, thng 5 nm 2009

Ngi vitSinh vin. Phm Trn Nguyt Tho


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PHN M U

1. L DO CHN TI

Nh bit, vic nghin cu qu trnh ng trong t nhin cng nh trong

x hi thng dn n vic kho st mt hay nhiu phng trnh o hm ring

bng vic nh lng ha cc c trng ca i tng nghin cu bng cc i

lng ton hc. Nhng ta cng d nhn thy rng cc quy lut t nhin thng

dn n cc h thc phi tuyn gia cc tham bin nn cn phi xt phng trnh

vi phn phi tuyn. Tuy nhin, khi xut hin nhng kh khn ton hc thc s.

Bi vy, khi xy dng m hnh ton hc chng ta buc phi bt tnh chnh xc v

b qua nhng phn thm phi tuyn b hoc chuyn sang tuyn tnh ha trong mtln cn ca nghim cho bng cch a bi ton v bi ton tuyn tnh. Vn

cha , gii bi ton ny ta li c nhng s thay i nht nh i vi gi

thit ca bi ton tng ng nghim ca n cng c nhng thay i nht nh.

Khi , vic tm nghim c in ca bi ton mi vn cn rt phc tp, v th,

utin ngi ta xy dng nghim suy rng ca n, sau thit lp trn ca

chng v chng minh n l nghim c in ca bi ton. Ni nh vy thy

rng, khng gian nghim ca bi ton c gii c nhiu thay i so vikhng gian nghim ca bi ton thc t ban u. V vy, vic chn cc khng

gian hm cho nghim ca cc bi ton c mt vai tr quan trng m bo tnh

t ng ca bi ton. Mt khng gian phim hm tuyn tnh c s dng rng

ri trong l thuyt phng trnh o hm ring l khng gian Sobolev.

C phn yu thch ton hc ng dng v c s hng dn gi ca

thy Phm Gia Khnh hon thnh lun vn tt nghip kha hc cng nh

bc u lm quen vi Phng trnh o hm ring hin i em quyt nh

chn ti Mt s vn v khng gian Sobolev .

2. MC CH NGHIN CU

Mc ch nghin cu nhm nm c cc nh ngha, nh l, tnh cht

lin quan n khng gian Sobolev. c bit quan trng v cng l mc tiu chnh

l xy dng h thng v d minh ha v gii cc bi tp c lin quan n n.


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Qua , gip cng c cc kin thc c hc trong sut 4 nm i hc nh:

gii tch 1, 2, khng gian tp, o v tch phn Lebesgue, gii tch hm

3. PHNG PHP NGHIN CU

Qu trnh lm lun vn s dng kt hp nhiu phng php nghin

cu, nhng ch yu l phng php tng kt kinh nghim.

C th, kt hp phng php tng hp, so snh, phn tch, nhn xt trong

qu trnh nghin cu l thuyt. u tin, sau khi tm c ngun ti liu tham

kho th tng hp cc kin thc trong vi cc kin thc sn c. Sau , tin

hnh so snh, phn tch chng, chn ra nhng kin thc trng tm, ng ghi nh,

t a ra nhng nhn xt ring. Cui cng, tng hp, trnh by li theo hiu

mt cch r rng.

Phng php tng hp, phn tch, so snh, nh gi cng c kt hp sdng trong gii v sp xp cc bi tp. Sau khi tng hp cc bi tp t cc ngun

t liu khc nhau, s tin hnh phn tch, so snh, chn lc cc bi tp hay ph

hp vi ni dung l thuyt sp xp chng theo mt trnh t hp l. Cui cng, tin

hnh phn tch, nh gi gii mt cch chi tit, trnh by mt cch r rang.

4. NI DUNG LUN VN

Ni dung ca lun vn gm 2 chng.

Chng 1. Gm 4 , trong 1. Khng gianp

L ,2. Bin i Fourier,3. Hm suy rng, 4. Khng gian Sobolev. y, ch tp trung gii thiu mt

s nh ngha, nh l, tnh cht v cc v d c lin quan m khng quan tm n

vic chng minh cc nh l, tnh cht .

Chng 2. Gii chi tit v sp xp mt cch tng i hp l cc bi tp

c lin quan n phn l thuyt gii thiu chng 1.


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PHN NI DUNG

CHNG 1

KIN THC CHUN B1. Khng gian pL

1.1 Khng gian pL

Cho ,, S l mt khng gian o, trong l mt tp con m ca

khng gian Euclide n chiu Rn, Sl -i s trn tp o c Lebesgue v l

o Lebesgue. Cho p1 , ta nh ngha khng gian pL nh sau

Vi p1 , ta nh ngha

pL { ff : l hm o c v xdxfp

}

v

p

pp

p

pdfxdxff

/1/1

Vi p , ta nh ngha

L { ff : l hm o c v kxf hu khp ni 0, k }

v

f inf { KxfK :0 hu khp ni}

Ch . Ni kxf hu khp ni tng ng vi ni rng

0: Kxfx .

Nu gf, l hai hm o c tha xgxf hu khp ni th f v g

c xem l ging nhau. Do , 0p

f khi v ch khi 0xf hu khp ni,

vi p1 .

Cho p1 , ch s q tha 111

qp

c gi l s m lin hp cap.

Ta thy, 1p th q . Ngc li, p th 1q .


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1.2 Mt s nh lv bt ng thc

1.2.1B

Cho a, b l hai s thc khng m,p, q l cp s m lin hp. Khi ,

q

b

p

aab

qp

.

1.2.2Bt ng thc Hoder.Nu qp LgvLf th

1Lfg v qp gffg 1

Du = xy ra khi v ch khi BA, R+ sao cho qp xgBxfA .

1.2.3Bt ng thc Minkowski.Nu pLgf , th

pppgfgf , vi p1 .

Du = xy ra khi v ch khi BA, R+, 022 BA sao cho BgAf .

1.2.4nh l. pL l khng gian Banach.

1.2.5nh l. pL l khng gian phn x, vi p1 .

1.3 Tch chp

Cho 1, Lgf , tch chp ca f v gc nh ngha l

dyygyxfxgf

1.4 Gi ca hm

1.4.1nh ngha. Cho f l mt hm lin tc trn Rn. Gi ca f , k hiu

l supp f , l bao ng ca tp 0: xfx .

K hiu cC (Rn) l k hiu tp tt c cc hm lin tc vi gi compact.

cC (Rn) thng c vit l D (Rn).

1.4.2 V d

Cho :f R Rc xc nh

0,0

0,2/1

x

xexf

x

Khi , Cf .


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Cho :f Rn Rcxc nh

ax

axexf

xaa

,0

,))/((222

, vi 2212

... nxxx

Khi , xf D (Rn

) v supp f axxaB :),0( . Cho 0 v nh ngha /xx n , vi 1L (Rn), 11 v

0x khi 11

. Tht vy,

1/ nnn RRn

Rdyydxxdxx , vi /xy .

Cho 0 v nh ngha /xCx n , vi nR dxxC 1 v: Rn R c cho bi hm

1,0

1,))1/(1( 2

x

xex

x

Khi , x D (Rn) v supp ),0( B

2. Bin i Fourier

2.1 K hiu

nxxxx ,...,, 21 (Rn),

n

j

jjxx

1

. , vi ,x (Rn).

nndxdxdxxdm ...

2

1212/

o c Lebesgue trn Rn.

yxfxfy , vi y thay i trn Rn,

/1

xfxfn

, 0 ,

;11

ffy

11

ff .

Cho 1, Lgf (Rn), tch chp nR dyygyxfxgf ,111 gfgf .

a ch s jn a,,...,, 21 N,

n

j

ja

1

. Cho Rn,

n

n

...21 21 vij

jx

, v nnD

...21 21 .


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Cho 1Lf (Rn), bin i Fourier ca f c nh ngha l nR

xi xdmexff .

Vi mi Rn, xiex l mt hm c trng trong Rn.

2.2 Tnh cht c bn

1Lf (Rn),1

ff

.

1Lf (Rn), 0 Cf ( Rn). 1, Lgf (Rn), gfgf ^ . fef yiy ^ , fxfe xi ^ 00 v ff . Nu 1Lf (Rn) v 1Lfj ( Rn) th fif jj ^ .

Nu 1Lf (Rn) v 1LfD (Rn), k , th .^ fifD

Nu 1Lf (Rn) v 1Lxfxj (Rn) th f kh vi n j v ^ xfixf jj

Nu 1Lf (Rn), 1Lfx (Rn) v fD tn ti, th

^ xfixfD

2.3 V d

(Gauss) 2/2xex ; 2/2 e ; 2/12

n

ij

jxx .

(Poisson)

2/121/

n

nxCx , vi nC lm cho 11 th

e .

(Fejer)

n

j

j

j

nx

xCxK

1 2

2

2/

2/sin;

n

jK

11 .

(de la Vallie Pousin) (cho 1n ) xKxKxV 22 . Khi ,

2,0

2,2

,1

V .


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2.4 nh l o cabin iFourier

Nu 1Lf (Rn) v 1 Lf (Rn) th dmefxf xiRn

hu khp ni.

2.5 nh l Plancherel

Nu21

LLf (R

n

) th2

Lf (R

n

), 22

ff v nh x

:F 21 LL (Rn) 2L (Rn) c cho bi fFf c thc trin thnh mt ng

c 2L (Rn) 2L (Rn).

2.6 Khng gian Schwartz S

Khng gian Schwartz S l khng gian ca cc hm tiu chun m bt bin

i vi cc php ton bin i, php nhn cn bng, m rng, nhn bi hm c

trng, tch nh x, php tnh tch phn v php bin i Fourier. Khng gian

Schwartz S c m t

S { C (Rn): ,,supnR

xDxx

}

y , l a ch s.

Mt vi ch

Ch SD nn S tr mt trong pL (Rn), p1 . Mt hm

2x

ex

, 0 thuc Snhng khng thucD.

Cho mt a thc P v S , SxxP v SDP . S khi v ch khi vi mi s nguyn 0k v vi mi a ch s ta

c xDx k 21 gii ni.

l song nh t S vo S. Khi , ta c cc kt qui. ^ xixD .

ii. ^ iD .


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2.7 Hm suy rng iu ha

2.7.1nh ngha. Khng gian tp i ngu 'S ca khng gian

Schwartz Sc gi l khng gian ca nhng hm suy rng iu ha.

2.7.2 V d

Cho pLf (Rn), p1 , nh ngha CSTf : c xc nh bi nRf dxxxffT ,

Khi , 'ppf

fT do fT l lin tc.

Nu M (Rn) (Khng gian ca cc o gii ni thng thng, ingu ca 0C (R

n)), xt

n

R

xdxT

Khi 'ST .

Chof l mt hm o c trn Rn sao cho vi mi s nguyn khngm kta c pk Lfx 21 (Rn), vi p1 . Khi ,

nRf dxxxfT

xc nh mt hm trong 'S , do

fT

nRkk

dxxxxfx

22

11

nn hm cho l hm iu ha.

Nu l mt o thng thng trn Rn sao cho Mx k 21 (Rn), theo cch xc nh trn 'ST . o cho c gi l o iu ha.

2.7.3nh l. Mt hm tuyn tnhL trn Sl mt hm iu ha nu v ch

nu tn ti mt hng s 0

C v s nguynl,

msao cho

SCLml

,,

, .


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2.7.4 Ton t trong S . Cho T S.

Php tnh tin. Nu hRn, nh ngha STT hh , th'STh .

Php nhn vi mt phn t ca S. Cho S , nh ngha TT . Khi , 'ST . Nu P l mt a thc trn Rn, PT c nh

ngha ging nh trn cng l mt hm iu ha.

Php phn x. ~~ TT . Khi , ST~ . Php tnh vi phn. Cho mt a ch s , nh ngha

SDTTD ,1

(Cng thc trn cho ta mt php tnh tch phn). Do , 'STD .

Tch chp. Cho S nh ngha TT . Khi ,'ST .

Lc , ta xt hm xTxF . Khi CF (Rn) v . TdxxTdxxTdxxxF nnn R xR xR

iu ny ch ra rng, tch chp vi mt phn t ca Sl mt qu trnh trn. Vi

mi hm iu ha T, th CT .

Bin i Fourier.nh ngha , TT S . Khi , ' ST .Kt hp php bin i Fourier v php vi tch phn, ta c

i. TxiTD

ii. TiTD

2.7.5 V d. Cho T tha 0 T . Khi

xx

)0('

j

0,1 x = x

jj , S , vi phn l mt trng hp c bit ca tch chp.


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3. Hm suy rng

3.1 Khng gian cc hm chun D

3.1.1 nh ngha. Mt hm tiu chun nxxxx ,...,, 21 trn Rn l

mt hm kh vi v hn trn

v trit tiu bn ngoi min gii hn, min giihn c th ph thuc vo hm tiu chun. Khng gian tt c cc hm tiu chun

trn c k hiu l D .

3.1.2 V d.

Cho : R R c xc nh bi

1,0

1,1/12

x

xex

x

D dng kim tra l mt hm v hn kh vi, tr trng hp 1x . V l hm chn, ta ch cn kim tra tnh kh vi ti 1x .

Ta c,

0lim 11

1

2

x

xe

Suy ra lin tc ti 1x

Hn na,

0lim 11

11

1

22

x

xx

em

Do , tt c o hm ca bng 0 ti 1x

Mt hm tiu chun i xng cu trn Rnc cho bi

1,0

1,1/12

x

xex

x

Trong , x l khong cch t tm nx.

3.1.4 Mt s tnh cht

Nu D21 , th Dcc 2211 vi mi s thc 21 cvc . Nu thuc D v a kh vi v hn trn th .a cng thuc

D .


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Nu thuc D th mi o hm ring ca cng thuc D . Cho hm nh trong v d hm tiu chun i xng cu, khi

0

xxcng l mt hm tiu chun trn Rn trit tiu ngoi hnh cu tmx0 bn

knh .

Cho Dxxxm ,...,, 21 (R

m) v Dxx nm ,...,1 (Rn-m). Nu

nxxx ,...,,. 21 mxxx ,...,, 21 . nm xx ,...,1 th D. (Rn).

3.2nh ngha v dy rng

Chng ta ni mt dy Dm l mt dy rng trong D nu

0m , trong D tn ti mt tp con compact c nh K sao cho

supp Km vi tt c m, m v tt c o hm ca n u hi t u n 0

trn K.

3.3 nh ngha v hm suy rng

Mt hm tuyn tnh T trn D c gi l mt hm suy rng trn

nu 0mT vi mi dy rng m trong . Khng gian cc hm suy rng

c k hiu l 'D .

3.4 nh l. Cho f l mt hm kh tch a phng trn mt tpcon m Rn. nh ngha

dxxxfTf

Khi , fT thuc 'D .

Nhn xt. Cho pLf , 1p . Khi , fT 'D .

V d

Hm suy rng Dirac.Cho x Rn, nh ngha

Dxx , (Rn)

D dng chng minh rng 'Dx (Rn).

Trng hp 0 c gi l hm suy rng Dirac.


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Cho Tc nh ngha bi 0nnT , D (R), n=1,2,...

Khi , 'DTn (R).

Trng hp 1n , 1T c gi l hm suy rng lng cc.

Nhn xt. khng c sinh ra bi bt c hm kh tch a phng no.

Tht vy, nu tn ti mt hm kh tch a phng f sao cho fT , khi

00 BR B dxxfdxxxfdxxxfn

vi D sao cho supp 0 B , 10 , 1 trn 02/B . Do ,

0 khi 0 .

Mt khc, 1 , vi mi . V vy, 0 khi 0 l muthun.

3.5 Tnh cht ca hm suy rng

Nhc li. Cho n

,...,, 21 , trong i , ni ,...,2,1 , l cc s

nguyn dng, khi c gi l mt a ch s.

K hiu mt s php ton lin quan n a ch s

n

ii

1

ni i1 !! xxx ni i i ,1 Rn

Cho hai a ch s nn ,...,,,,...,, 2121 . Khi , khi

v ch khiii , vi mi n,2,1,i .

l mt a ch s, ta nh ngha php ton vi phn D l

n

nxxD

...11

.

Tnh cht 1. Cho 'DT , vi l tp m con ca Rn, v a ch s .

Khi ,

DDTTD ,1 .


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Tnh cht 2. Cho 'DT , Rn l tp m v C . Khi ,

DTT , .

Tnh cht 3. Cho 'DT , R l mt tp con m, D , v mt

a ch s , ta c cng thc Leibniz

TDDTD

!!

!.

V d. Cho :H R R l hm Heaviside c cho bi

0,0

0,1

x

xxH

Ta c

0''' 0 dxxTT HH

Khi , HT' .

Nhn xt. T v d trn, r rng '' ff TT .

3.6 Tch chp ca hm suy rng

Cho hm u bt k trn Rn v x Rn, k hiu

)( xyuyux v yuyu

.

Suy ra

uuxx v yxyx .

Vi 'DT (Rn), D (Rn), v x Rn, nh ngha

xx

TT

D thy 'DTx (Rn).

3.6.1nh ngha. Cho 'DT (Rn), v D (Rn), :T Rn Rnc cho

bi

x

TxT , vi mi x Rn.


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3.6.2 nh ngha. Cho ', DST (Rn) Ta nh ngha hm suy rng ST

trn D (Rn) l

0

STST , D (Rn).

nh ngha 3.6.2 tng ng vi iu kin

STST , vi mi D (Rn).

4. Khng gian Sobolev

4.1 Khng gian Sobolev

Cho l mt tp m con ca Rn c bin l . Ta bt u vi nh

ngha.

4.1.1 nh ngha. Cho s nguyn m>0 v p1 . Khng gian

Sobolev c nh ngha

mLuDLuW pppm ),()()(, pmW , l tp hp tt c cc hm thuc )(pL c o hm suy rng n m

cng thuc )(pL .

Ta c )(D , khng gian ca tt c cc hm kh vi v hn vi gi compact

trong , th tr mt trong )(pL , vi p1 . Nu )(D th )(DD ,

vi mi a ch s . Nh vy,

)()()( , ppm LWD , vi p1 .

)(, pmW l mt khng gian vct.

Trn )(, pmW ta trang b mt chun,,

.pm

, nh sau

Vi p1 , ta nh ngha

p

m

p

Lpmp

uD

/1

0)(,,

.

.

Vi p , ta nh ngha

)(0,,max

Lmm

uDu

.


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Trng hp c bit 2p , ta k hiu )()(2, mm HW , cho )( mHu ,

khi

,2,, mmuu

Vi 0

m , ta c )()(

,0 ppLW , chun trn

p

L ca hm )( p

Lu ck hiu l

)(pLu .

Khng gian )(mH c mt php ton nhn trong t nhin c nh

ngha

m

m vuDDvu

,),( , vi )(,

mHvu

Php ton nhn trong ny sinh ra,

.m

.

Trong trng hp Rn

, (mH Rn

)

c mt s m t khc qua bin iFourier.

Cho (1Lu Rn),

nR

x dxxfeu )()( 2

l s bin i Fourier ca u.

Ch . (1L Rn) (2L Rn) th tr mt trong (2L Rn), nhng hm trong

(2L Rn) c s bin i Fourier thch hp, nh l Plancherel khng nh rng

)R()R( n2n2

LLuu .

Cho mHu (Rn), theo nh ngha ta c (2LuD Rn), vi mi

m , nh vy )( uD c xc nh tt. Hn na, Ta c uiuD )2()^( .

Do , ( 2Lu Rn), vi mi m .

Ngc li, nu (2Lu Rn) sao cho ( 2Lu Rn), vi mi m ,

th (2LuD Rn), vi mi m . V th (mHu Rn).

4.1.2 B . Tn ti hng s 00 21 MvM ch ph thuc m v n sao

cho vi mi Rn,

m

m

m MM )1()1(2

2

22

1


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T b , chng ta c nh ngha ca (mH Rn).

4.1.3 nh ngha

(mH Rn)

)()(1)( 2

2/22 nm

nRLuRLu

Kt hp vi chun

222

)()()1( uu m

RRHnnm .

T nh ngha trn cho php chng ta nh ngha (sH Rn), vi mi 0s .

4.1.4 nh ngha. Cho 0s , nh ngha

(sH Rn)

)()(1)( 2

2/22 ns

nRLuRLu .

Kt hp vi chun222

)()()1( uu s

RRHnns .

4.1.5 nh l. Vi mi p, p1 , )(, pmW l mt khng gian Banach.

* Xt khng gian tch: )1((,...1 nLLL ppnp ln)

Vi chunp

n

i

p

Lipuu

/11

1)(

, vi 111 ),...,(

np

nLuuu

Khi , nh x 11

, ,...,,)(

np

n

pm Lxu

xuuWu l mt php

ng c. Ta c mt s tnh cht

)(, pmW l khng gian phn x, vi p1 . )(, pmW l khng gian tch c, vi p1 . )(mH l khng gian Hilbert tch c, vi p1 .4.1.6 nh ngha. Cho p1 , t )(,0

pmW bng bao ng ca

)(D trong )(, pmW .

)(,0 pmW l mt khng gian con ng ca )(, pmW .

Phn t ca )(,0 pmW gn ging trong khng gian nh chun

)(, pmW bng nhng hm thuc C c gi compact trn .


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)(,0 pm

W l khng gian con thc s ca )(, pmW , tr trng hp Rn.

4.1.7 nh l. Cho p1 , khi (,1 pW Rn) = (,10pW Rn).

4.1.8 nh l. Cho p1 , vi mi s nguyn 0m th

(,pmW Rn) = (,0 pmW Rn).

Trng hp c bit mH (Rn)= mH0 (Rn).

Ta c th ni rng pL l mt tp gm cc lp hm. Nh vy, khi ni u

l mt hm lin tc trong pL ngha l ta ang ni ti mt lp hm m c i

din l hm u lin tc.

Kt qu sau c trng cho pW ,1 khi I R l mt khong m.

4.1.9 nh l. Cho I R l mt khong m, nu IWu p,1 th u l hmlin tc tuyt i.

4.1.10 Ch . Cho I R l mt khong m gii ni, v d )1,0(I . Khi

, nu IWu p,1 th ta c th vit

dttuuxux

0 '0

Nh vy,

q

IL

qpx px

xuxuxdttuxudttuxuu p/1/1/1

00''')0(

Ly tch phn trn 1,0 , ta c

IpILIL

q

ILucuucdxxudxxuu ppp ,,111

1

0

/11

0'')0(

trong 1c khng ph thuc u. Cng nh vy ta c

IpIpIpIpucuucuuxu

,,13,,0,,12,,0'')0()(

Trong ,2

c v 3c c lp vi u.

4.1.11 Ch . Khng gian ln hn cha khng gian ca nhng hm trn

hn.


19/64

- 19 -

Ly B(0,1) 1:,,1

,1 Ip

p uIWu l hnh cu n v trong IW p,1 . Khi

, nh x )(: ,1 ICIWi p lin tc. Do , B(0,1)=i(B(0,1)) l mt tp gii ni

u trong )(IC .

Mt khc, cho Iyx , , ta c

q

Ip

q

ILyxuyxuyuxu p

/1

,,1

/1'

suy ra B(0,1) lin tc u trong IC , t nh l Ascoli-Arzela suy ra B(0,1) l tp

compact tng i trong IC . Hay ni cch khc, )(: ,1 ICIWi p l mt ton

t compact.

Trn khng gian )(, pmW , ta nh ngha na chun

p

ma

p

Lpm puDu

/1

,,

c xem l o hm cao nht ca khng gian nh chun pL .

4.1.12B (Bt ng thc Poincare). Cho l mt tp m gii ni

trong Rn. Khi , tn ti mt s nguyn dng pCC , sao cho

,,1 pLuCu p , )(

,10

pWu .

,,1 puCu nh ngha mt chun trn )(

,1

0p

W tng ng vi chun

,,1.

p. T 0

,,1

pu theo bt ng thc Poincare suy ra 0u . Do , n l mt

chun.

Ta c,

p

p

p

p

p

p

p

L

p

L

p

puCuCuuuDu pp

,,1,,1,,11

,,1)1(

v

p

p

p

Lp

p uuDu p

,,11,,1 .

T hai bt ng thc trn, ta c

p

p

pp

p

p

puCuu

,,1

/1

,,1,,1)1( .


20/64

- 20 -

4.1.13 V d. Bt ng thcPoincare khng ng vi min khng gii ni.

V d, nu ly Rn v D (Rn), xc nh bi

2x,0

1,1 xx , 10

t kxxk

/ , th

0

/1

1)(,,1

p

p

RLkRpk np

n D

, khi k .

Trong khi )),0(()(

kBnp RLk , khi k .

4.2 Khng gian i ngu ca khng gian Sobolev

phn ny ta xt khng gian sobolev ca s nguyn m cng nh phns.

4.2.1 nh ngha. Cho p1 , q l s m lin hp ca p. Khng gian

i ngu ca khng gian )(,0 pmW , vi m l s nguyn ln hn 1, c k hiu l

)(, qmW .

Nh vy, mH l khng gian i ngu ca mH0

.

4.2.2 nh l. Cho q

WF,1

, khi , tn ti q

n Lfff ,...,, 10 sao cho

p

i

n

i

iWv

x

vfvfvF ,10

10 , (2.1)

V

q

LiniqfF

0,,1max .

Khi l tp gii ni, ta c th gi s rng 00 f .

Gi s nh l trn l ng. V D tr mt trong

pW ,1

0, hm tuyn

tnh th xc nh duy nht nn n c nh trong D .

Cho D , (2.1) c vit li nh sau

n

i i

i

i

n

i

ix

ff

xffF

10

10


21/64

- 21 -

Nh vy, F c th c xc nh vi hm suy rng

n

i i

i

x

ff

10 . Mt

hm trn pW ,1 th c xc nh vi mt hm suy rng, l o hm suy rng

ca mt phn t trong qL . Do , khng gian i ngu ca pW ,10 c k

hiu l qW ,1 .

nh l trn cng ng vi khng gian i ngu ca pW ,1 (tr trng

hp ta khng gi s 00 f ngay c khi gii ni), nhng s xc nh vi hm

suy rng th khng th. Tht vy, khng gian i ngu ca pW ,1 cng bao

gm s m rng ca hm suy rng trn pW ,1 , nhng s m rng ny khng

duy nht.

Cho pnm / , khi CW pm,

Do , gi tr ca im c nh ngha tt.

Nu 0x , D , th

00 xx

v

,,00 pmLx Cx .

Cho khng gian nh chun pmW , ,0x

lin tc trn D v tnh lin

tc c thc trin n pmW ,0 . Suy ra qmx W,

0 , vi pnm / .

Vi mi min xc nh , hm suy rng Dirac thuc khng gian Sobolev

ca mt s m ln no .

By gi, ta nh ngha khng gian Sobolev cho mt s thc s bt k.

nh ngha. Cho p1 , th

ppnsp

ps Lyx

yuxuLuW

/, : .

Cho ms , 10,0 m ,

mWuDWuW ppmps ,: ,,, .


22/64

- 22 -

psW ,0 l bao ng ca D trong ps

W, v qsW ,0 l i ngu ca

psW ,0 .

Khi Rn v 0s ,

s

H (Rn

)={ 2Lu (Rn

) : 22/2

1 Lus

(Rn

)}v

2/1

22

)(1

nns R

s

RHduu

Khng gian i ngu ca sH (Rn) l sH (Rn), khi s>0.

4.2.3 nh l. Cho 0s , khi

sH (Rn) 'Su (Rn): 22/2 1 Lus (Rn)}

4.2.4 Ch . Nu l hm suy rng Dirac th

1

Do ,

dxxnR 0

V

sH (Rn) 22/21 Ls (Rn).

iu ny ng cho 2/ns , v tch phn

0 2

1

1dr

r

rs

n

ch hu hn khi

2/ns .

Trng hp c bit, sH (R), 2/1s .


23/64

- 23 -

CHNG 2

BI TP

Bi 1

a. Chng minh rng nu gf, l hm lin tc c gi l tp compact th

supp gf supp f + supp g .

Gii

a. GiA, B ln lt l gi ca f v g. Gi s

BzAyzyBAx ,:

Xt

.dyygyxfxgf D thy tch phn khc khng ch khi By v Ayx . Nhng nu

BAx th cy vx-y ln lt khng thuc voB vA. V vy,

0 xgfBAx

V cA vBu l compact nn A+B l compact. V vy,

supp gf supp f + supp g .

Bi 2. Cho pLfp

1 . Chng minh rng

1,:sup qq

pgLgfgdf .

Gii

Trng hp 1. p1 . Theo bt ng thc Holder ta c

qpgffgd

Do , v phi lun ln hn hoc bng v tri. Trng hp ng thc xy ra, gi

s 0p

f . ( 0p

f ng thc hin nhin ng). t fffgpp

psgn

11 . Khi

, qLg v 1q

g . Hn na,

p

pp

pfdfffgd

1


24/64

- 24 -

Trng hp 2. 1p . Khi ,

gffgd 1 . Nh vy mt chiu ca

bt ng thc xy ra. Gi s 01

f . t fg sgn . Khi , 1

g v

1ffgd .

Trng hp 3. p . Mt chiu ca bt ng thc hin nhin ng nh

trc. Gi s 0

f . Cho

f0 . Chn mt tp o c A sao cho

A0 v xf , Ax . nh ngha

AfA

g

.sgn1

trong , A

l k hiu hm c trng caA. Khi , 1Lg v 11

g . Khi ,

A dfAfgd

1

Do , phn trn l ng cho mi (

f0 ).

Bi 3. Cho hm baLf loc ;1 , ta ni hm baLg loc ;

1 l o hm suy rng

ca f nu

b

a

b

adxgdxf ' vi mi baCc ;

K hiudx

dfgfg ;' . Hy chng minh cc tnh cht sau cao hm

suy rnga.o hm suy rng l duy nht hu khp ni

b. ''' 2121 ffff

c. '' cfcf .

Gii

a. Vi mi baCc ; , gi s c 21, gg tha

b

a

b

adxgdxf 1' v

b

a

b

adxgdxf 2'

Suy ra

021 b

adxgg , .

Do , 21 gg hu khp ni. Hay o hm suy rng l duy nht hu khp

ni.


25/64

- 25 -

b. Vi mi baCc ; , ta c

b

a

b

a

b

a

b

a

b

a

b

adxffdxfdxfdxfdxfdxff ''''''' 21212121

M

b

a

b

adxffdxff '' 2121

Suy ra

b

a

b

adxffdxff ''' 2121

Vy ''' 2121 ffff .

c. Vi mi baCc ; , ta c

b

a

b

a

b

a

b

adxcfdxfcdxfcdxcf ''''

M

b

a

b

adxcfdxcf ''

Suy ra

b

a

b

adxcfdxcf ''

Vy '' cfcf .

Bi 4. Gi s x l hm Heaviside

0,0

0,1

x

xx

v cc hm suy rngx

1v x c nh ngha: vi mi hm th Dx (R)

x

dxx

xx

x 0lim,

1v 0, xx

Hy chng minh cc ng thc sau

a. xxdxd

b. xxdxd 1

ln

c. xxdx

dsgn , trong xxx sgn

d. xxdx

d , trong xxx .


26/64

- 26 -

Gii

Gi s Dx (R) tha supp aa,

a. Ta c

xx ,' xx ',

a

a

dxxx '

a

dxx0

'

ax0

0 a 0 xx ,

Vy xx ' .

b. Ta c

xx ,ln' xx ',ln a

adxxx 'ln

a

adxxxdxxx

'ln'lnlim

0

aa

aadx

x

xxxdx

x

xxx

lnlnlim

0

a

adx

x

xdx

x

x

lnlim

0

x

xdx

x

x

x

,

1lim

0

Vyx

x1

ln' .

c. Ta c

xx ,' xx ', a

adxxx '

a

adxxxdxxx

''lim

0

aa

aadxxxxdxxxx

0lim

a

adxxdxxaaa

0lim

a

adxxdxx

11lim

0

a

adxxxsigndxxxsign

0lim

a

adxxxsign xxsign ,

Vy xsignx ' .


27/64

- 27 -

d. '' xxx xxxx '' xxx ' xxx

Ta chng minh 0xx .

Tht vy,

xxx , xxx ,

t xxx 1 . Khi , x1 cng l mt hm th v

xxx , xxx , xx 1, 01 00 0

Vy xx ' .

Bi 5. Cho Cf v g l hm lin tc vi gi compact. Chng minh

rng Cgf .

Gii

chng minh Cgf ta ch cn chng minh

g

x

f

x

gf

ii

, ni 1 .

u tin ta chng minh gf l hm lin tc. Gi s supp Kg . Chn

mt tp compact Csao cho CKx va CKhx , vi h nh. Khi , f

l lin tc u trn C. Khi ,

K

dyygyxfyhxfxgfhxgf .

Cho 0 . Chn 0 sao cho CKhx v

yxfyhxf vi h . Do , xgfhxgf 1

g hay

gf l hm lin tc. Xt 0,...,0,1,...,0ie , trong 1 xut hin v tr th i.

Khi ,

h

xgfhexgfi

K i dyygyxfheyxfh

1

Ki

i

dyygheyxx

f (1)

vi mi , 10 . Vix

f

l lin tc trn C v b gii ni nn theo nh l v s

hi tb chn ca Lebesgue v phi ca (1) hi t v xgx

f

i

.


28/64

- 28 -

V vy

gx

f

x

gf

ii

.

Bi 6. Cho p1 , pLgLf ,1 , chng minh rng

pLgf vpp

gfgf1

.

Gii

+ Vi 1p hoc p ta d dng c c iu cn chng minh.

+ Vi p1 , cho qLh . Xt hm s

)()()(, xhygyxfyx

Khi hm ny l o c. Hn na,

dxxhdttxgtfdxdyxhygyxf )()()()()()(

dtdxtxgxhtf )()()(

.1

fhgqp

Do , nh x hgfh l mt hm tuyn tnh lin tc trnqL . Do

, pLgf v .1 pp

gfgf

Bi 7. Cho hm suy rng bt k 'DT (Rn), cc hm tiu chun

D21,, (Rn), x Rnl im bt k v l a ch s bt k. Khi ,

a. xxx TTT

b. DTTDTD

c. 2121 TT

d. Nu 0T vi mi D (Rn) th 0T .

Gii

a. Cho y bt k thuc Rn

xyTyTx

xyT

yxT

yx

T yTx

Ngoi ra, ta c

yTTTyTxxyxyx


29/64

- 29 -

b. Ta s chng minh cho trng hp 0,...,1,...,0,0 ie vi 1 v tr

th i. Trng hp bt k c chng minh tng t.

xx

Th

T

xTh

xTxTh

xTxTh

hexTxTh

xTx

i

x

he

xh

heh

heh

heh

ih

i

i

ii

i

0

00

00

lim

1lim1lim

1lim

1lim

Ngoi ra,

xxT

xT

Th

xTh

xTx

i

x

i

xheh

heh

iii

1

lim1

lim00

c. Ta phi chng minh rng

xTxT 2121

vi mi x Rn. Do

0 TxT x

T cu a suy ra

00 2121 TT

khi

2121 0 TT

m

dyyxdyyx

dyyyxxx

py

Ry

R

n

n

2sup2121

212121

Tch phn cui trn tp compact supp

2 c th c xem nh gii hn

khi 0 ca tng Riemann

p

ppx 21 trong tng c m rng trn

tt c im nt tch phn trn Rn.


30/64

- 30 -

Do ,

021lim

p

ppx 21

trong D (Rn). Do ,

0

lim0

21sup 21

210

2121

2

TdyyyT

pTTT

p

p

p

d. Ta phi chng minh rng 0T , vi mi D (Rn). Ta c

0

TT

M D (Rn) suy ra

D (Rn). Do ,

0

T

Suy ra

00

T

Vy 0T .

Bi 8. Cho 'T (Rn) v (Rn), ta nh ngha

x

TxT .

Trong , l mt khng gian tp c sinh ra bi mt dng hi t c

bit trn C v ' l lp cc hm suy rng vi gi compact. Chng minh

rng

a. TTT xxx b. DTTDTD , vi l a ch s bt k.

c. Cho T ' (Rn) v D (Rn) th DT (Rn) v

T = T = T .


31/64

- 31 -

Gii

ng thc a v b c th c chng minh tng t ng thc a v b trong

bi 7. chng minh cu c,t K supp(T) v H supp(). Khi , KvHl

compact. T nh ngha,

x

TxT

Ta c supp

x Hx , 0 xT nu KHx suy ra

HKx . Do ,

supp HKT supp(T)+supp()

V supp T l mt tp ng trong tp compact K+H nn n l tp

compact. iu ny chng minh rng DT (Rn). Tr li vn , chng

minh cu c. ta cn chng minh ti hm gc. Xt D0 (Rn) sao cho

HK

0

trong DHK

(Rn) l mt hm ngng ca K+H.

Khi ,

00

000

0

TT

dyyyTdyyyTT HKHK (1)

Khi , xt Hh v Kk th

khkhkkhh

00

Do , 0 TT trn H. T ,

0

0

00

TdyyyT

dyyyTT

H

H

(2)

T (1) v (2) ta c

T = T (3)


32/64

- 32 -

Ly 1 bt k thuc D (Rn). Khi , s dng cc tnh cht ca tch chp v

(3), ta c

1111 TTTT

S dng cu dcabi 7, ta c TT .

Bi 9.Cho 'DTi (Rn), 3,2,1i . Khi ,

a. Nu 1T hoc 2T thuc ' (Rn) th 1221 TTTT .

b. Nu t nht hai trong ba iT ' (Rn), 3,2,1i th

321321 TTTTTT

c. Vi bt k a ch s ta c 212121 TDTTTDTTD .

Gii

a. Cho D21 , (Rn), ta c

1221212121212121 TTTTTTTT

Nu 1T ' (Rn) th ta c th s dng cu c ca bi 8 v nu 2T ' (R

n)

th ta c th s dng cu dcabi 7 c 12212121 TTTT (1)

Ngoi ra, t (1) ta cng c

21121212 TTTT (2)

Do tch chp c tnh giao hon nn v phi ca (1) bng v phi ca (2).

Do , ta c

2121 TT 2112 TT

Do ,

2121 TT 2112 TT

v 2 l ty , ta c

121 TT 112 TT


33/64

- 33 -

Ta li c, 1 l ty , ta c

21 TT 12 TT .

b. Nu 21,TT ' (Rn) th 21 TT ' (R

n). Do , nu t nht c 2

'DTi (Rn), 3,2,1i , thuc vo ' (Rn) th c 321 TTT v 321 TTT c

xc nh.

Xt 3T ' (Rn), ta c

321321321 TTTTTTTTT

v

321321321 TTTTTTTTT

v DT 3 (Rn). Do l ty , ta chng minh c b.

By gi, nu 3T ' (Rn) th c 21,TT ' (R

n). Do ,

123231321 TTTTTTTTT

V 1T ' (Rn) nn

321123123 TTTTTTTTT

c. Cho D (Rn) v a ch s , ta c

2121212121 TDTTDTDTTDTTTTD

do l ty , ta chng minh c c.

Bi 10. Cho 'DT (Rn), khi ,

TTT .

Mt cch tng qut, vi a ch s bt k ta c

TDTD

Gii

Cho D (Rn). Khi ,

xxxxx

0 .


34/64

- 34 -

Do . Ngoi ra,

TTTT

Do D (Rn) l ty nn

TTT .Vi a ch s bt k ta c

TDDTTDTD .

Bi 11. Cho 1L (Rn), 0 v 11

. Chng minh rng 0 , c

nh ngha l /xxn , vi 0 , l mt xp x ng nht thc, ngha l

ff ,1Lf (Rn), khi 0 .

GiiDo

dyxfyxfyxfxfnR

,

Ta c

dxdyxfyxfyffn n

R Rn /

11

yny yn

dyyf

ffy ./2

/1 1

1

Cho 0 , chn 0 sao cho vi mi ta c 1ff .

Bi 12. Cho p1 . Chng minh rng D (Rn) tr mt trong pL .

Gii

Xt S l lp cc hm kh tch n gin (Tc l 0: xx ). Khi

, trit tiu bn ngoi tp c o v hn nn pL .

Xt pLf , 0f . Khi , tn ti mt dy n S sao cho nn f ,0

hi t theo o v f. Hn na,pp

nff . Do , p

n Lf v

0 pnf , khi n . Suy ra, tp cc hm n gin khng m tr mt trong

pL .


35/64

- 35 -

Cho S v 0 . Theo nh l Lusin, tn ti mt hm cCg (Rn) sao

cho g hu khp ni, tr mt tp c o b hn tha

g . Do ,

pp

p

pdgg

/1/1

2 .

Suy racC (R

n) tr mt trong pL . ChocCf (R

n), d thy

KBfpfp 1;0sup*sup

Trong , K l tp compact v ff * . Do ,

0,0*** Kffdffffpp

K

p

p.

Bi 13.

a.i vi khng gian S(Rn) ton t Fourier F: S(Rn) S(Rn) l mt ng

cu tp. Hy chng minh cc cng thc sau

i) gfgf ii) ff ..

iii) dxgfgdxf

iv)

dxgfdgfn

2

v) gfgf . vi) gfgf n 2.

b. Cc cng thc i), ii), iii), iv), v), vi) trong cu a s nh th no, nu

dng php bin i Fourier theo cng thc

?2 .2/ n

R

ixn dxxfef

c. Chng minh cc cng thc v), vi) trong cu a i vi trng hp

1Lf (Rn) hoc 2Lf (Rn) v g S.

Gii

a.

i) Ta c

gfdxexgdxexfdxexgfgfnnn

R

xi

R

xi

R

xi

Vy gfgf .

ii) Ta c

fdxexfdxexff xiR

xi

Rnn

...

Vy ff .. .


36/64

- 36 -

iii) Ta c

dxxgxf dxdxgfe xi

t x . Suy ra ddx Khi ,

dxxgxf

dxdgxfe

xi

dxdgexfxi

dxxgxf Vy dxgfgdxf

.

iv) Ta c

dgf dxdgxfe xi dxdgexf xi

dxdgexf xi

dxxgxf

n2

Suy ra

dxgfdgfn

2 .

v) Ta c

gf dxxgfe xi dydxyxfeyg xi

t yxX . Suy ra yXx v dxdX . Khi ,

gf dydxyxfeyg xi dydXXfeyg yXi

dydXXfeyge Xiyi dyfyge yi

dyygef yi gf

Vy gfgf . .

vi) Ta c

gfdyygyxf

dydfeygdydygfe

ddyygefedgfexgxf

nn

yxinyxin

yinxixi

22

22

2.

Suy ra gfgfn

2.

.b. Ta c

nR

xin dxxfef 2/2

Suy ra

nR

xin dfexf 2 2/


37/64

- 37 -

i) Ta c

gfdxexgdxexf

dxexgdxexfdxexgfgf

nn

nnn

R

xin

R

xin

R

xi

R

xin

R

xin

22

22

2/2/

2/2/

Vy gfgf

.ii) Ta c

fdxexfdxexff xiR

nxi

R

n

nn

.2.2. 2/2/

Vy ff .. .

iii) Ta c

dxxgxf dxdxgfe xin

2/2

t x . Suy ra ddx Khi ,

dxxgxf dxdgxfe xin 2/2

dxdgexf xin

2/2 dxxgxf

Vy dxgfgdxf

.

iv) Ta c

dgf dxdgxfe xin 2 2/ dxdgexf xin

2 2/

dxdgexf xin

2 2/ dxxgxf

Suy ra

dxgfdgf .

v) Ta c

gf

dxxgfe xin

2/2 dxdyygyxfe xin

2/2

dydxyxfeyg xin

2/2

t yxX . Suy ra yXx v dxdX . Khi , dydXXfeygedydXXfeyggf XinyiyXin

2/2/ 22

dyfyge yi dyygef yi

gfn 2 2/

Vy gfgf n .2 2/ .


38/64

- 38 -

vi) Ta c

xgxf . dgfe xin 2/2

ddyygefe yinxin 22 2/2/

dydygfeyxin

2

dydfeyg yxinn 2/2/ 22

dyygyxfn 2 2/

gfn 2 2/

Suy ra gfgf n 2. 2/ .

c.Trc tin, ta chng minh vi 1Lf (Rn) hoc 2Lf (Rn) v g S th

gf u c xc nh tt. Tht vy,

+ Vi 1Lf (Rn), Ta c

dyygyxfxgf dyygyxf

Suy ra 1Lgf v

1L

gf dxxgf dxdyygyxf

dydxyxfyg 11 . LL gf

+ Vi 2Lf (Rn), Ta c

dyygyxfxgf dyygyxf

Suy ra 2Lgf

V

2L

gf 2/12dxxgf 2/12

dxdyygyxf

2/1

22.

dxdyygdyyxf

2/122 dydxyxfgL 22 . LL gf

Nh vy vi 1Lf (Rn) hoc 2Lf (Rn) v g S th gf u c xc

nh tt.


39/64

- 39 -

By gi, ta s chng minh cc cng thc gfgf . ,

gfgf n 2. cho trng hp 1Lf (Rn). Do S tr mt trong 1L (Rn) nn

f 1L (Rn) th kf S sao cho ffk . Tc l,

Nk 0,0 sao cho 0kk th 1Lk ff .

+ Khi , ta c

dxxgfxgfgfgf kLk 1

dxdyygyxfyxfk

dydxyxfyxfyg k

dyygffLk 1 01 Lg

v

dffff kL

k 1 ddxxfxfe kxi

ddxxfxfe kxi

ddxxfxfk

dffLk

1 0 d

Do ,

gfgf k

klim gfgfk

k..lim

Vy gfgf . .

V

gfgf

kklim. gfgf n

k

n

k22lim

Vy gfgf n 2. .

+ By gi, ta s chng minh cc cng thc gfgf . ,

gfgf n 2. cho trng hp 2Lf (Rn). Do S tr mt trong 2L (Rn) nn

f 2L (Rn) th kf S sao cho ffk . Tc l,

Nk 0,0 sao cho 0kk th 2Lk ff .


40/64

- 40 -

+ Khi , ta c

2/12

2

dxxgfxgfgfgf kLk

2/12

dxdyygyxfyxfk

2/122 dxdyygdyyxfyxfk

2/122 dydxyxfyxfg kL

22LkL

ffg 02 L

g

V

2/12

2

dffff k

Lk

2/12

ddxxfxfek

xi

2/12

ddxxfxfek

xi 2/1

2

ddxxfxfk

02 Lk

ff

T ,

gfgfgfgfn

nn

n..limlim

.

Vy gfgf . .

V

gfgf k

klim. gfgf nk

n

k22lim

.

Vy gfgf n 2. .

Bi 14.

a. Vi f S(Rn), hy chng minh cng thc

fifD

b. Hy thit lp cng thc i vi fD vi f S(Rn)

c.Dng nh ngha ca php bin i Fourier ca f S nh sau

,,, ff S(Rn)

tm fDfD , , vi f S.


41/64

- 41 -

Gii

a. Trc tin ta chng minh rng vi mi i ta c fifx

i

i

.

Tht vy,

dxxfx

efx

i

xi

i

dxxfe

x

xi

i

dxxfei xii fi i

Khi , vi mi i ta c

f

x i

i

i

dxxf

xe

i

i

i

xi

dxxfe

x

xi

ii

i

i

1

dxxfeixi

i

i

fii

i

Suy ra

fifD .

b.Trc tin ta i tm cng thc tnh

fi

, vi mi i.

Ta c

dxxfedxxfef si

i

si

ii

dxxf

edxxfe

i

si

i

si

dxxfixe ixi xfixi

Khi , vi mi i ta c

dxxfedxxfef si

i

si

ii

i

i

i

i

i

i

dxxfedxxfei

i

i

i

i

si

i

si

dxxfixe iixi xfix ii

Suy ra

xfixfD .


42/64

- 42 -

c.

+ Tm fD

DffD ,1, Df,1 ,1 if

,fi , fi

Vy fifD .

+ Tm fD

,fD ,fD ,1 Df ixf,1

ixf ,1 ,fix

Vy fD fix .

Bi 15. Gi s 1Lf (R). Chng minh rng

a. Lf (R), 1 LLff

,

b.f lin tc u trn R,

c. Nu ta cng c 1' Lf (R), chng minh rng 0 f khi .

Khng c thm gi thit 1' Lf (R), iu khng nh trn c cn ng khng?

Gii

a. Ta c

1LRR

xi

R

xi fdxxfdxxfedxxfef

T nh ngha chun ca L (R) suy ra1

LL

ff

.

b. Ta c

0' ''

R

xixi

R

xixidxxfeedxxfeeff

, khi '

Suy ra f lin tc u trn R.

c.Do 1' Lf (R) nn 1' Lf (R) v R df .

Ta c

fif '


43/64

- 43 -

Do ,

'' fi

i

ff

Cho , ta c

0'

limlim

fif

Ta khng cn n iu kin 1' Lf (R). Tht vy, ta c th chn

1', Lgg (R). (Khi , 0 g khi .) sao cho 1

1 LL

gfgf . Khi

,

khig

ggfggfggffL

,0

1

Bi 16. Tm bini Fourier ca cc hm s sau

a.

,1

22x

xf R, 0 b. 0,2

1 4/2 aea

xf ax

c.

,

sin1

x

xxf R d.

khcx

cxbexf

x

,0

,

e.

khcx

cxbxf

,0

,1 f. xaa;

g. ,xa a>0 hng s, x R.

Gii

a.

,1

22x

xf R, 0 . Ta c

dx

x

ef

xi

22

Xt

xexg .

Ta c

220

0

0

0 2

ie

i

edxedxedxeeg

xixixixxixxix


44/64

- 44 -

M

degxg xi

2

1

Suy ra

dee xix

222

21

t y , ta c

dye

ye

iyxx

22

2

2

1

t

x

xy, ta c

de

xe ix2222

1

Suy ra

edex

ix

22

1.

b. 0,2

1 4/2 aea

xf ax

.

Ta c

dxea

e

dxea

dxeea

f

a

xaia

aaxxiaiaxxi

22

2222

2

4/4/

2

2

1

2

1

ta

dxdtai

a

xt

22 .

Khi ,

Iedteefa

ta

2

2

2

By gi ta i tnhI. Ta c

0

2

0

2

0 0

2

22

2222

2

1

.

rr

utut

edrdre

dtdueduedteI


45/64

- 45 -

Suy ra

I

Do ,

2

2

a

a

e

e

f

Vy 2 aef .

c.

,

sin1

x

xxf R. Ta c

0 0

00

0

0

sinsin1

sincos21sin1

sinsin1sin1

dxx

xdx

x

x

dxx

xxdx

x

xee

dxx

xedx

x

xedx

x

xef

xixi

xixixi

By gi ta tnh tch phn

0

sindx

x

xI

Theo bin i Laplace ta c

0

1

dtex

xt

Do ,

dtxdxedxdtexIxtxt

0 00 0sinsin

Xt

0sin xdxeK xt

t

t

evdxedv

xdxcoxduxuxt

xt

sin

Khi ,

00

0

sinxdxcoxe

txdxcoxe

tt

xeK xtxt

xt


46/64

- 46 -

t

t

evdxedv

xdxduxu

xtxt

sincos

Suy ra

0

2

2 sin xdxett

Kxt

Do ,

22t

K

Nn

dttI

022

t duutgdttgut 21 . Khi

21

12/

022

22

utg

duutgI

T ,

1sinsin1

0 0

dxx

xdx

x

xf

Vy 1 f .

d.

khcx

cxbexf

x

,0

,.

Ta c

i

eee

i

dxedxeedxxfef

bicic

b

xi

c

b

xic

b

xxixi

1

Vy

i

eef

bici

.


47/64

- 47 -

e.

khcx

cxbxf

,0

,1.

Ta c

i

eee

idxedxxfef

cibic

b

xic

b

xixi

1

Vy

i

eef

cibi .

f. Ta c

adxx

dxxixdxedxe

a

a

a

a

a

xixi

aaaa

sin2cos2

sincos

0

;;

Vy aaa sin2 ; .

g.

ax

axxa

,0

,1 , vi a l hng s ln hn 0.

Ta c

a

i

ee

i

edxedxxae

aiaia

a

xia

a

xixi sin2

Vy

asin2

.

Bi 17. Tnh ff vi

a. 1;0xf b. x

xxf

sin

Gii

Ta p dng tnh cht 2. fffff

a. Ta c

111 101

01,0

ixixixixi e

ie

idxedxexdxexff

Suy ra

222 1

1

iefff

Vy 22 11

ieff .


48/64

- 48 -

b. Ta c

aaa

sin2 ;

Do

defxf xi

21

Nn

dea

xxi

aa

sin1

,

Suy ra

dea

x xiaa

sin1,

t dxdx . Khi , ta c

dxe

x

xa xiaa

sin1,

Ta c

,

sin

dxexx

dxexff xixi

Do

,

22

f

Vy

,

2

ff .

Bi 18. Cho 2

xexg

. Tnh gg .

Gii

Ta c

degdeggxgg

xixi

22

1*

2

1*

M

4/2

eg Nn

2/22

eg


49/64

- 49 -

Suy ra

221

.42

2

2

2

2

2.

2

12

1

2

2.

2

1*

x

xixi edeedteexgg

Vy 22

22*

x

exgg .

Bi 19. Cho 0,2

22

xxf . Vi 0, , tnh ff .

Gii

Ta c

e

x22

1

Theo nh l o ca php bin i Fourier ta c

22

4

2

14

22.22

1

.2

1

2

1

xdee

deedeee

deffdeffxff

ixb

ixbix

ixix

Vy 22

4

xxff .

Bi 20. Tm hmf tha

22221

bxayx

dyyf

, ba 0 .

Gii

t

22

1

axxg

a

Khi

a

ae

ag


50/64

- 50 -

V

22 ayx

dyyfdyyfyxgxfg

aa

Biu thc cho tr thnh

xgxfg ba

Suy ra

ba gfg

Hay

ba gfg .

T

ba

ebfea

Ta c

abeb

af

M

2221

2

12

1

abxbabadee

abbaba

deeb

adefxf

xiab

xiabxi

Vy

22 abxbaba

xf

.

Bi 21. Cho 1Lf (R). t

dxxxfa

cos1

v

dxxxfb

sin1

Nu f lin tc, chng minh rng

dxbxaxf

0

sincos

p dng tm hm f tha:

2,0

21,2

10,1

sin0

dxxxf .


51/64

- 51 -

Gii

Ta c

dxbxa

ddyyyfxdyyyfx

dydyxyfdyxyf

dydyxyf

dydyxyfidyxyf

dydyxiyxyfdydeyf

dedyeyfdefxf

yxi

xiyixi

0

0

0 0

0

sincos

sinsincoscos1

sinsincoscos1

cos1

sincos2

1

sincos21

21

2

12

1

p dng, t

0,

0,

xxf

xxfxF

Theo chng minh trn ta c

dxbxaxF

0

sincos

Ta c

2,0

21,4

10,2

sin2

sin1

0cos1

cos1

cos1

0

0

0

dxxxfdxxxFb

dxxxfdxxxfdxxxFa


52/64

- 52 -

Khi

x

xx

x

x

x

x

dxdxdxbxFxfx

4cos2cos8cos4cos2

sin4

sin2

sin

22

1

1

0

2

1

1

000

Vy 0,4cos2cos8 2

xx

xxxf

.

Bi 22. p dng nh l Plancherel tnh cc tch phn sau

a.

dx

x

x2

sin

b.

221 x

dx

c.

dx

x

xxx6

2sincos.

Gii

nh l Plancherel cho ta22

22

2LL

ff .

a. Ta c

sin2 1;1

Theo nh l Plancherel ta c

2

1;1

2

1;1 22 2LL

Hay

dxxd

2

1;1

2

1;1 2

Do

1

1

2

42sin2

dxd

Suy ra

dxx

xI

2sin


53/64

- 53 -

b.t

21

1

xxf

Khi

ef


22

222

LLff

Hay

dxxfdf

22

2

Do

dxx

de

22

2

1

12

Suy ra

211

0

2

22

dedxx

.

c. t

1;12 1 xxf

Khi

1

0

21

1

2 cos12sincos1 dxxxdxxixxdxexff xi

t

xvxdxdv

xdxduxu

sincos

212

. Ta c

1

0

1

0

1

0

2

sin

4sin

2

sin

12

xdxxdx

x

x

x

xf

t

xvxdxdv

dxduxu

cossin

.


54/64

- 54 -

Ta c

3

1

0

1

0

sincos4coscos4

dx

xxxf


dxxfdf

22

2

Hay

dxxd

1

1

226

2

12sincos16

Suy ra

.15

21

3

2

5

1

43

2

54

124

14

sincos

1

0

35

1

0

241

0

22

6

2

xxx

dxxxdxxd

Bi 23. Gi s AdfnR . Chng minh rng

ddbaf 22 LL baA .

Gii

Ta c

ddbaf ddbaf

ddaffb 21

.22/12/1

ddafdfb 2/12

ddafbA 2/122/1

2

1.22/12

2/1 ddafbA

2/1222/1 ddafdbA

2/122/1 2 ddafbAL

2/122/1 2 ddfabA L 22 LL baA .


55/64

- 55 -

Bi 24. Chng minh rng vi mi hm s x S(Rn), vi 12/ nk ta

lun c

xn

R

max nk RHC. .

Gii

Vi 12/ nk th

dnR

k22/2

1 .

Do , 22/2 1 Lk (Rn). Suy ra kH (Rn), v

2/1

221

dnnk R

k

RH

Khi , ta c

x

nR

xin

de

2

nR

xin

de

2

nR

n

d

2

21

.22/22/2 1.12

nR

kkn

d

2/1

222/1

21.12

nnR

k

R

kn

dd

nk RHC. , vi mi x Rn.

Suy ra xn

R

max nk RHC. .

Bi 25. Xt dy cc hm s xxfm , R

mx

mxm

xfm

2

1,0

2

1,

Chng minh rng xfm

l dy c bn trong H-1(R).

Gii

Ta c

m

mxm

dxmdxxixmdxemdxxfef

m

m

m

m

m

m

m

m

xi

Rm

xi

m

2

2sin

sin

cossincos

2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1


56/64

- 56 -

Do

m

mf

m

2

2sin

nn 1lim

m

mf , tc l

0,02 n

sao cho 0nm

th 21

mf v

0',02n

sao cho 0'nn th 2

1

n

f

Khi 00 ';max,0 nnN sao cho Nm v Nn th

nmnmnm ffffff 1111

T ,

Nnm

dd

ff

dff

ffff

RR

nm

R

nm

RLnmRHnm

,,011

11

2/1

2

2/12

2

2/12

2

2/12

21

Vy mf l dy Cauchy trong H-1(R).

Bi 26. Cho xf H-s(Rn), x Hs+1(Rn). Chng minh cng thc tch

phn tng phn

nn

Rj

Rj

dxx

xxfdxx

x

xf .

Gii

Cch 1. Chng minh bngphng php quy np.

+ Ta chng minh cng thc ng vi 1n . Tht vy,

R dxxxf ' R xfdx RR dxxxfxxf ' R dxxxf '

+ Gi s cng thc ng vi kn . Tc l

kk

Rk

jR

k

j

dxdxdxx

xxfdxdxdxx

x

xf...... 2121

, vi .,...,1 kj


57/64

- 57 -

Ta phi chng minh cng thc ng vi 1 kn . Tc l phi chng minh

11 121121 ...... kk

Rk

jR

k

j

dxdxdxx

xxfdxdxdxx

x

xf , vi .1,...,1 kj

i) Trng hp .,...,1 kj

121121 ......1

kR R

k

jR

k

j

dxdxdxdxxx

xfdxdxdxx

x

xfkk

121 ...

k

R Rk

j

dxdxdxdxx

xxf

k

1 121 ...k

Rkk

j

dxdxdxdxx

xxf

ii) Trng hp 1 kj

1

...... 2111

1211

k kR Rk

Rk

k

k

k

dxdxdxdxxx

xfdxdxdxx

x

xf

k

R Rk

kRkk

dxdxdxdxx

xxfxxf

k...211

111

1 1211

...k

Rk

k

dxdxdxx

xxf

Suy ra cng thc ng vi 1 kn .

Vy ta chng minh c cng thc tch phn tng phn.Cch 2

Xt 0,...,0,1,0,...,0je , 1 xut hin v tr th j.

t xfhexff j v xhex j . Ta c

xfxfxfxf xfxfxf

ejhxfxfxf

ejhxfxfxf

fejhxfxfxf

fxxfejhxhexfxf j


58/64

- 58 -

Khi

nn R

jhR

j

dxxxfhexfh

dxxx

xf

1lim

0 nR h dxxfh

1lim

0

n

R

j

h

dxfxxfejhxhexfxf

h

1

lim0

nRj

hhdxxxfejhxhexf

hxf

h

1lim

1lim

00

nRjj

dxx

xxf

x

xxf

nnR

jR

j

dxx

xxfdx

x

xxf

1 111...nn R njjRjjR j dxdxdxdxxxfdxx

x

xf

n

Rj

dxx

xxf

Vy

nn

Rj

Rj

dxx

xxfdxx

x

xf .

Bi 27. Cho x Hs(Rn), x Hs(Rn), xx Hs(Rn),

j

xxx Hs-1(Rn),

jx

xx Hs-1(Rn). Chng minh cng thc Leibnitz

jjjx

xxx

x

xxx

x

Gii

Xt 0,...,0,1,0,...,0je , 1 xut hin v tr th j.

t xhex j v xhex j

Suy ra xhex j v xhex j

Khi

hexhex jj xx

xxxx


59/64

- 59 -

Ta c

xxxj

xxhexhexh

jjh

1lim

0

xxhh

1lim

0

xxhexxxhexh

jjh

1lim

0

xxhexh

xxhexh

jh

jh

1lim

1lim

00

xhex

hxxxhex

hj

hj

h

1lim

1lim

00

jj

x

xxx

x

x

Ch . 01

lim0

hh

. Tht vy,

xhexxhexhh

jjhh

1lim

1lim

00

xhex

hxhex

hh

jjh

11

lim0

xhex

hxhex

hh

jh

jhh

1

lim1

limlim000

00

jjx

x

x

x

Vy ta chng minh c cng thc Leibnitz

jjjx

xxx

x

xxx

x

.

Bi 28. Ta nh ngha hm suy rng xax , Rn, 0a bng ng

thc sau

ax xdxxax ,

trong xd l phn t din tch mt cu bn knh a trong Rn.

a. Tm bin i Fourier ca hm ax b. Chng minh rng ax 12/ nH (Rn).


60/64

- 60 -

Gii

a. Ta thy ax S(Rn) v supp ax axx , . Nh vy,

ax c gi compact. Do ,

xi

eaxax

,

ax xxi

de

axxdxixcox sin

ax xax x dxidxcox sin

V xsin l hm l theo x , v mt cu axx , i xng qua tm, nn

ta c

0sin ax xdx

Do ,

ax ax xdxcox .

b. Do 1xcox v mt cu axx , l tp compact, nn tn ti hng s

C sao cho

Cax

Khi , ta c

daxn

R

n 212/21

dCnR

n 12/22 1

Suy ra 22/12/21 Laxn (Rn)

Vy ax 12/ nH (Rn).

Bi 29. Cho X l mt khng gian Banach. Ta ni hnh cun v

1: xXxS l li ngt nu yx , 1 yx th , 10 ta c

Syx 1 . Xt xem hnh cu n v trong QC , QL1 , QL2 c li ngt

khng, Q l mt min b chn trong Rn.

Gii

+ Ta chng minh hnh cu n v trong QC khng li ngt.

Tht vy, chn 0tx , 1tx , 1max txx

QtQC, 1ty , vi mi

t QC .


61/64

- 61 -

Khi , vi mi tha 10 , ta c

tytxyxQtQC

1max1

1max txQt

11

Suy ra yx 1 khng thuc hnh cu n v trong QC .

Hay hnh cu n v trong QC khng li ngt.

+ Ta cng chng minh c hnh cu n v trong QL1 khng li ngt.

Tht vy, chn 0tx , QL dttxx 11 , Qty1

, vi mi Qt .

Khi , vi mi tha 10 , ta c

QL dttytxyx 11 1

Q dtQtx

1

1

QQ dtQdttx1

1 11

Suy ra yx 1 khng thuc hnh cu n v trong QL1 .

Hay hnh cu n v trong QL1 khng li ngt.

+ Hnh cu n v trong QL2 li ngt.

Tht vy, yx , 122 LL

yx v tha 10 , ta c

2/1

211 2

QL dttytxyx

2/12

1

Q dttytx

2/1

2222 112

Q dttytytxtx

2/1

2222 112

QQ Q dttydttytxdttx

2/1

2222 112

LQL ydttytxx

2/122 22121LL

yx

1121 2/122

Suy ra yx 1 thuc hnh cu n v trong QL2 .

Vy hnh cu n v trong QL2 li ngt.


62/64

- 62 -

Bi 30. Cho f l mt hm kh tch a phng trn mt tp con m

Rn. nh ngha dxxxfTf . Chng minh rng fT thuc 'D .

Gii

D thy fT l mt hm tuyn tnh trn D . Ta chng minh n lin tc.Cho m l mt dy rng trong D . Xt K l tp compact c nh sao

cho supp Km , vi mi m. Khi ,

dxxfxT mKxmf max .

V m

l mt dy rng nn 0maxlim

xm

Kxm

Do 0mfT khi m .

Bi 31. Gi s dy ...,2,1, mxfm cc hm thuc QCk hi t yu trong

QL2 n mt hm f, cn dy ...,2,1, mfD m , vi kn ),,...,( 1 , b

chn trong QL2 , Q l mt min b chn trong Rn. Chng minh rng hm fc

o hm suy rng fD .

Gii

V m

fD b chn trong QL2 nn theo nh l Bolzano-Weierstrass tn

ti dy con kmfD ca mfD hi t trong QL2 . Gi s ufD Lmk 2

.

Vi mi hm mf QL2 xc nh mt hm suy rng QDT

mf' . Khi ,

mTT fL

fm,

2

Tht vy,

QLdxxxfxfdxxxfxfTTQ

mQ

mffm

2,0

Tng t, ta c mTTu

L

fDkm

,2

.

Theo cng thc Ostrogradski, ta c

fDf

k

fm

k

fDmu

TDTDTTTkm

kkm

k

1lim1lim

Do ufD theo ngha ca o hm suy rng.

Vy tn ti fD .


63/64

- 63 -

PHN KT LUN

Phng trnh o hm ring ra i vo khong th k th XVII do nhu cu

ca c hc v cc ngnh khoa hc khc. N ngy cng c vai tr quan trng,

c ng dng rng ri trong khoa hc v cng ngh. Ngy nay, phng trnh

o hm ring tr thnh mt b mn ton hc c bn va mang tnh l thuyt cao

va mang tnh ng dng rng. Trc s pht trin nh v bo ca khoa hc cng

ngh, chc chn rngphng trnh o hm ring s cn pht trin mnh m hn

na trong tng lai, m ra mt con ng cho nhng ai yu thch nghin cu

ton hc ng dng. Trong qu trnh hc tp, c thy c gii thiu, em cm

thy rt c hng th vi mn hc ny. Cho nn, khi c lm lun vn em xinc nghin cu vphng trnh o hm ring. Tuy nhin, khi bt tay vo lm

em mi thy rng, tuy trn th gii phng trnh o hm ring v ang pht

trin mnh, nhng nc ta vn cn rt t sch ni v ti ny, nu c th cng

i su nghin cu lm sng t l thuyt, cn bi tp ch l gi n gin. V th,

c s gi ca thy hng dn, bc u lm quen vi phng trnh o

hm ring, trong lun vn ca mnh em tm hiu v khng gian Sobolev -

c nh ton hc Sobolev S.L gii thiu vo gia th k XX, n nhanh chngtr thnh mt cng c t lc trong vic gii phng trnh o hm ring, do ,

n c nhiu nh ton hc khc tip tc m rng v pht trin nhm nghin cu

cc phng trnh o hm ring ngy cng kh khn phc tp - Nhng khc

ch, trong lun vn ca mnh, em i su chn lc mt h thng cc v d minh

ha, gii v sp xp tng i hp l cc bi tp lin quan n khng gian

Sobolev. Chnh v th, c th xem cun lun vn nh mt t liu, hnh trang cho

em sau ny nu c iu kin s tip tc nghin cu v phng trnh o hm

ring.

Do y l ln u tin thc hin nghin cu khoa hc, nn em khng th

trnh khi nhng thiu st nht nh. Em rt mong nhn c s quan tm, ng

gp kin ca qu thy c v cc bn lun vn ca em c hon chnh hn.


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TI LIU THAM KHO

[1] Nguyn Minh Chng, Phng trnh o hm ring, Nh xut bn Gio dc, 2000.[2] D. Bahuguna V. Raghavendra B.V. Rathish Kumar, Sobolev Space and

Applications, Alpha science.

[3] Nguyn Minh Chng, L thuyt phng trnh o hm ring, Nh xut bn Khoa

hc v k thut, 1995.

[4] ng nh ng, Nhp mn gii tch, Nh xut bn Gio dc, 1998.

[5] ng nh ng, Bin i tch phn, Nh xut bn Gio dc, 2001.

[6] Nguyn Xun Lim, Gii tch hm, Nh xut bn gio dc.

[7] GS.TSKH Phan Quc Khnh, Ton chuyn , Nh xut bn i hc Quc gia TP.H Ch Minh, 2000.

[8] GS.TSKH Nguyn Duy Tin, Bi ging gii tch, i hc Nng, 2004.

[9] M.A Trn Th Thanh Thy, Gio trnh Tp i cng, i hc Cn Th, 2004.

[10] M.A Trn Th Thanh Thy, Gio trnh o v tch phn Lebesgue, i hc Cn

Th, 2007.

Một số vấn đề về không gian Sobolev

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