các kết quả chuẩn bị các không gian hàm

trình bày về các kết quả chuẩn bị các không gian hàm

CHUaNG 2

CAC KET QUA CHUAN BJ CAC KHONG GIAN HAM

11.1 CAC KHONG GIAN HAM

D~t 0 = (0,1), ta bo qua dinh nghia cac khong gian ham thong

dl;lng: em (0), LP(0), Hm (0), Wm,p(0).

Vai m6i ham v E eo (0) ta dinh nghia II vII nhusau

(1

)

1/ 2

(2.1) Ilvll=llvllH=lfrv2(r)dr .

Ta dinh nghiaH la d~y du h6a cua khong gian eo (0) d6i vai

chua'n 11.11.

Tuong tlf, vai m6i ham v E el (0) ta dinh nghia 11.11 v nhusau

(2.2)

IIvIlv=(llvI12+II/IIT'

va dinh nghia V la d~y du h6a cua khong gian el (0) d6i vai chua'n 11.11 v

Chli Y r~ng cac chua'n 11.11 va 11.11 v l~n luQt duQc sinh ra tu cac

tich vo huang

(2.3)

1

(u,v) = fru(r)v(r)dr,

0

1

(2.4) (u,v)+(ul,v/)= fr[u(r)v(r)+ul(r)/(r)] dr.

Khi d6, ta d~ dang chung minh dng H, V la cac khong gian

Hilbert

B6 d~ 2.1 V tru m~t trang H vdi phep nhung lien tf:lc.

Chung minh Hitin nhien ding II v II ~ II v IIv ' 'v'vE V, do do phep

nhung tu V v~lOH la lien t1;1C.M~t khac C](0) c V va tru m~t

trong H, do do V tru m~t trong H.

B6 d~ sail cho mQt s6 danh gia thuong sa d1;1ng.

B6 d~ 2.2 Vdi mQi v E Cl (0), &> 0 va r E [0,1], ta co:

(2.5)

(2.6)

(2.7)

(2.8)

IIvll2 ~11/112 +v2(l),

Iv(l) I~vGllvIIv'

Frlv(r)I~21Ivllv'

v2(l)~&llv/r +(2+ :)llvI12.

Chung minh.

0 ~ r2 < r ~ 1, ta co

IIvl12= frv2(r)dr=~v2(l)- fr2v(r)/(r)dr

]

~ ~v2(l) + Sri v(r)v/ (r) I dr

~~v2(l)+llvllll/112 .

<;~ v2(1) + ~(1IvI12+llvf}

Suy ra IIvl12~v2(l)+llv/r.

Do do (2.5) dU<;1c chung minh.

Nghi~m l(li (2.6).

]

Ta co v2(1)=f(r2v2 (r))/ dr

0

1 1

= 2 frv2(r)dr + 2 fr2v(r)v/ (r)dr

~21IvIl2+2I1vllll/11

~211v112+lIv112+II/f

~3(lIvIl2+11/f)=3I1vll~

V~y I v(1) I~ J311 v IIv va (2.6) ducjc chung minh.

Nghi~m l(;li (2.7).

Ta co

2 fs v(s)/ (s )ds = fs d(v2(s))

1

=v2(1) - rv2(r) - fv2(s)ds ~ v2(1) - rv2(r).

r

Suy fa

1

rv2(r) ~ v2(1) - 2 fsv(s)/ (s)ds

r

1

~ v2(1) + 2 Sri v(r)/ (r) I dr

0

~ v2 (1) + 211villi/II

~v2(1)+llvI12 +11/112

~31Ivll~ +llvll~ =4I1vll~.

V~y ~lv(r)I~21Ivllv'

Do do (2.7) ducjc chung minh.

Nghi~m l(;li (2.8).

Theo chung minh (2.6) ta co

V2(1):0; 211v112+211v1111/ II=211 v 112+ 2 J-)IvII.J&.1I/11

~(2+ ~)lIvIl2+c:II/f

Do do (2.8) du'<Jcchung minh.

B6 d~ 2.3

Ta dang nhtit H wJi HI (d6'i ngau cila H) Khi do ta co

V C H ==HI C vi, vdi cac phep nhung lien tl;lcva ndm tru mcJt.

Chung minh Tru'dc he't ta chung minh r~ng H nhung trong Vi

VI VcH,vdimQi wEH, anhxC;l Tw:V~R

1

xacdinhbdi vHTw(v)=(w,v)= frw(r)v(r)dr

latuySn tlnh lien t\lC tren V, tuc la Tw E vi.

Ta xet anh Xc;lT: H ~ Vi

wHT(w)=Tw'

Khi do ta co

(Tw,V)VI,v = (w, v), \iv E V, \iw E H.

Ta se chung minh r~ng toan tii'T thoa cac tinh cha't sail

(i) T : H ~ Vi la d(/fl anh,

(ii) IITwllvl~llwll, \iwEH,

(iii) T(H) = {Tw: w EH} la tru mcJttrang Vi.

Chung minh (i) D~ tha'yr~ng T tuySn tinh NSu Tw=0 thl

(W,V)=(TW,V)VIV=O,, \iVEV.

VI V tru ffi~ttrong H, nen ta co

(w,v)=O, \ivEH.

Do do w = O V~y T la don anh, nghla la, ffiQtphep nhung tu H

vao vi

Chung minh (ii) Ta co vdi inQi v EH,

IITwllvl = sup I(Tw,v)l= sup I(w,v) I

VEV,llvllv=l VEV,llvllv=l

::; sup IIWIIIIVII::; sup IIWllllVllv =IIWII.

VEV,IIv Ilv=l VEV,IIv Ilv=l

Chung minh (iii) Ta chung rninh ding rnQi phiS-rn ham tuye'n

tinh lien wc tren vi va trit$t lieu tren T(H) thl ding trit$t lieu

A

V I

tren

Coi L E (Vi ivai (L,Tw)VII Vi = 0,, \:fTwE T(H) Ta chung rninh

rang L =0, th~t v~y, do V phein X'.l,tuc la (VI)1 = V theo nghla

(*) \:fL E (Vi i, 31 E V: (L,Z)VII ,vi = (z,/)vl v' \:fz E Vi.,

Lffy z=Tw EVI ta co O=(L,Tw)VII,v1 =(w,/), \:fWEV.

Do V tru rn~t trong H lien ta co (w,/) = 0, \:fw E H.

V~y 1= O Theo (*) ta co

(L,Z)VII,v1 = (Z,l)VI,V =0, \:fzEVI.

V~y L trit$tlieu tren vi.

Chu thich 2.1 Tli b6 d~ 2.2, ta cling dung ky hit$u tich vo huang

(.,.) de chi c~p tich d6i ng~u giii'a V, Vi.

B6 d~ 2.4 Phep nhung V C. H ill compact.

Chung minh xern [5].

(

2

)

1/ 2

Chu thich 2.2 Tli b6 d~ 2.2 suy ra r~ng v2(1)+II/ II va

(2.9)

~llvll~ ::;11/112+v2(1)::;41Ivll~, \:fvEV

Th~y v~y, bfft d~ng thuc thu nhfft cua (2.9) co duQcla do

Ilvll~=11/112 +llvl12 ~II/f +II/f +V2(1)

~ 2(11 vf +vZ (1) ).

Bfft d~ng thuc con l~i cua (2.9) duQc suy ra tli

II/f +v2(1)~II/f +31Ivll~~41Ivll~

Ta chu yrang

(2.10) lirn Fr vCr) = 0, Vv E V.

r~O+

(xern [1] trang 128 )

Mi;it khac, do HI (&,1) C CO([&,1]), 0 < &< 1 va

(2.11) F&llvIIHI(8,1)~llvllv' VVEV.

Ta suy ra rang

(2.12) vi [8,1]E CO([&,1]), V &, 0 < &< 1.

Tli (2.10), (2.12) suy ra

(2.13) FrvECO([O,I]), VVEV.

11.2 KHONG GIAN HAM LP(O,T;X), 1~ p ~ 00

Cho X la khong gian Banach th\fc d6i vdi chuffn11.llx

Ta ky hi~u LP(O,T;X), 1~ p ~ 00, la khong gian cac lOp tudng

T

dudng chua ham u : (O,T) »X do duQc, sao cho ~Iu(t) II~ dt <00

o

vdi 1~ p <00

hay ~M>O:llu(t)llx ~M, a.e tE(O,T) vdi p=oo.

Ta dinh nghla chuffn trong LP(O,T;X), 1~ P ~ 00 nhusau:

T

J

1I P

IluIILP(O,r;X) ~ }IU(t)II;dt vdi bp<oo,

vaIIuII L'XJ (O ToX } = ess supIIu(t) Ilx

, , O<I<T

=inf{M>O:llu(t)llx ~M,a.e tE(O,T)} voi p=oo.

Khi do ta co cac b6 d~ sau day ma chung minh cua chung co th€

Hm thffy trong J.L.Lions [2].

B6 d~ 2.5 LP(O,T;X) III kh6ng gian Banach.

~+~= 1, 1< P < 00, (LP(O,T;X))I = L/ (O,T;XI) III d6i ngt1u

cila LP (O,T;X).

Ran nila, ne'u X phdn xg, tht LP(O,T;X) Gangphdn xg,.

I

B6 d~ 2.7 (L] (O,T;X)) = rfJ (O,T;XI).

Ran nila, cac kh6ng gian L\O,T;X), Loo(O,T;XI) kh6ng phdn xg,.

Chu thich 2.3 N€u X = LP (0) thl LP(O,T;X) = LP (0 x (O,T)).

Phan b6 co gia tri vectd.

Dinh nghia 2.1 Cho X la mQt khong gian Banach thlfc MQt anh

X(;ltuye'n tinh lien t\lC tit D«O,T)) vao X du<jcgQi la mQt phan b6

co gia tri trong X T~p cac phan b6 co gia tri trong X ky hi~u la

DI (O,T;X) = L(D(O,T);X) = {f: D(O,T) ~ X /f tuye'n tinh va

lien t\lC}.

Chu thich 2.4 Ta ky hi~u D(O,T) thay cho D((O,T)) ho~c

C:«O,T)) d€ chi khong gian cac ham s6thlfC kha vi vo h(;lnco

gia compact trong (O,T).

Djnh nghia 2.2 Cho f E D/ (O,T;X) Ta dinh nghla d(;loham

df theo nghla phan b6 cua f bdi c6ng thlic

dt

(2.14)

\ ~,rp) =- \1,~), Vrp eD(O,T).

Cae tinh eha't.

1/ Cho v E LP(O,T;X) Ta lam tu'ong ling vdi no bdi anh X(;l

~ :D(O,T) ~ X nhu'sau:

(2.15)

T

(~,cp)= fv(t)cp(t)dt, VcpED(O,T).

0

Ta co th~ nghi<$ml(;lir~ng ~ E D/ (O,T;X) Th~t v~y

i) Anh X(;l~ : D(O,T) ~ X hi~n nhien la tuy~n tinh

ii) Ta nghi<$ml(;lianh X(;l~ : D(O,T) ~ X lien We

Gia sa {cpj} c D(O,T) saG cho lim CPj= 0 trong D(O,T)ta co J-++OO

(

T

)

p

(

)

/

~ }I vet) II ~ dt ~9'j(t) IP dt

Do do limII(Tv,cpj)II =o V~y~ E D/ (O,T;X).

j.-++oo ~ O.

2/ Anh X(;lv H Tv la mOt don anh, tuy~n tinh tu LP(O,T;X) VaG

D/ (O,T;X), do do ta co th~ d6ng nha't Tv= v

Khi do ta co k~t qua san t!:)f.LI<H.TlfN~HEN

Bo de 2.8 (J.L.Lions [2])

000.333

LP (O,T;X) C DI (O,T;X) V(Jiphep nhung lien t1:lc.

D~o ham trong LP(O,T;X).

Do b6 d~ 2.8, vdi f E LP(O,T;X) ta co th~ coi f va do do df la

dt cac phffn til'cua DI (O,T;X) Ta co cac ke't qua sau.

B6 d~ 2.9 (J.L.Lions)

Neu f,fl EL1(O,T;X) thi f bang hdu het vdi mi}t ham lien t1:lCtu [O,T] vao X.

Chung minh b6 d~ 2.9 g6m nhi~u badc.

I

Bude 1 D~t H(t) = IfI(s)ds Khi do H :[O,T]~ X lien t\lc VI

0

fl ELI (O,T;X).

T /

f l h h- h"

radc et ta cling Illin rang - =- = t eo ng la p an

bO'.Th~t v~y, ta co

T

-,qJ = - H,- = - IH(t)-(t)dt

1

1

)

=- IfI(s)ds ~(t)dt =- IfI(s)ds I~(t)dt

(2.17)

T

= IfI (S)qJ(S) ds = (fl,qJ).

0

V~y dH = df = fl trong DI (O,T;X).

dt dt

Bude 2 Ta chung minh f =H + C thee nghla phan bO'

(C la hang )

Th~t v~y, gia sil' v=H - f ta co / =0 thee nghla phan be) (do

badc 1 ) Ta se chung minh rang v = C thee nghla phan bO'.

Ta co vi = 0 tadng dadng vdi

T

JV(S)q:/ (s)ds = 0, '\IepE D(O,T)

0

Cho epE D(O,T), ta co thS vi€t ep duoi d~ng ep=Aepo + Ij//, trong

do Ij/ E D(O,T), CPothoa Jepo(s)ds = 1, A = Jep(t)dt.

T

Th~t v~y, ta co J(ep(t)-Aepo(t))dt=O, nen nguyen ham cua

0

ep(t) - Arpo (t) tri<%ttieu t~i t= 0 se thuQcD(O,T)

t

ChQn Ij/(t)= J(ep(s)-Aepo(s))ds.

0

Trong (2.18) thay ep/ bdi Ij// ta thu dU<;1c

Jv(s) Ij// (s) ds = Jv(s)[ ep(s) - Aepo(s)] ds = 0,

hay

'\IepE D(O,T)

(2.19)

= Jep(s)ds Jv(t)epo(t)dt ,

'\IepE D(O,T).

D~t C=Jv(t)epo(t)dt ta suy ra tu (2.19) ding

0

T

J(v(s) - C) ep(s)ds= 0, '\IepE D(O,T).

0

V~y vet) = C = canst trong D/ (O,T;X).

Btioc 3.Ta sa dl;lngtinh cha't sail:

T Ne'u wELl (O,T;X) va JW(t)ep(t)dt = 0, '\IepE D(O,T) thi

0

DiSu nay co dU<;1cla do anh x~ W~ Tw tu LI(O,T;X) vao D/ (O,T;X) la don anh (tinh cha't 2/ d tren ) Ta suy ra ding

=

Tn cac bu'oc 1, 2, 3 d tren b6 d€ 2.9 dff du'Qc chung minh.

Tu'dng tlj ta co b6 d€ sau:

B6 d~ 2.10 Ne'u f,f/ E LP(O,T;X) thi f biing hdu he't wJi mQt

ham lien tl;lcta [O,T] vao X.

A? ;;; ;;; , ?

11.3 BO DE VE TINH COMPACT CUA J.L.LIONS

Cho 3 khong gian Banach X 0' Xl, X voi X a eX C Xl sao cho

(2.20)

(2.21)

Xo,XI la phan X£;l,

Phep nhung Xa~ X la compact

Voi 0 < T < +00, 1:::;; Pi :::;; +00, i = 0,1 Ta d~t

(2.22) W(O,T) = {v E LPo(O,T;Xo): / E LPI (O,T;XI)}

Ta trang bi cho W(O,T) bdi chufin

(2.23)

II vllw(o,T) =IIv IILPo(O,T;Xo) +II / IILPI (O,T;Xl)

Khi do W(O,T) la mQtkhong gian Banach

Hi€n nhien W(O,T) c LPo(O,T;X).

Ta cling co ke't qua sau dtiy lien quail de'n phep nhung compact

86 d~ 2.11 (B6 di vi tinh compact cila J.L.Lions).

Vcri giG thie't (2.20), (2.21) va ne'u 1< Pi < 00, i = 0, 1 thi phep nhung W(O,T) ~ LPo(O,T;X) la compact.

Chung minh C6 th€ Hm tha'y trong J.L Lions [2], trang 57

11.4.BO DE VE SV HQI TV YEU TRONG L q(Q)

B6 d~ 2.12

Cho Q la tcJp md, bi ch(in cila RN va Gm,G E Lq (Q), 1< q < +00, sao cho

Gm C, trong do C la hiing so'dQc lcJpvcri m va

Gm ~ G a.e (r,t) trong Q.

Khi do G m ~ G trong L q(Q) ye'u.

~ ~

11.5 BO DE GRONWALL

B6 d~ cu6i cung nay lien quail d€n ffiQt ba"t phuong trlnh tich

phan, no ra"t cfin thi€t cho vi~c danh gia tien nghi~ffi trong cae

chuang sau.

B6 d~ 3.13 (Bd d~ Gronwall ).

Gid siCI: [O,T]~ R la ham khd tich, khong am tren [O,T]va

t

thoa ba't ddng thac I(t):::; C1+ C2 fl(s)ds vdi hdu he't t E [O,T]

0

trong do Cj, C2 la cae hl1ng so' khong am.

Khi do l(t):::; CjeC2t vdi hdu he't t E [O,T].

Ta cling dung cae ky hi~u

u(t), ul (t) = ut(t), ur(t) = Vu(t), urr(t), lfin luQt d~ chI u(r,t),