Hàm Suy rộng CoLombeau 7

Luận văn Thạc Sĩ toán học-ngành Toán Giải Tích -Chuyên đề : Hàm Suy rộng CoLombeau

CHUaNG 3

MQT 56 KET QUA VE HAM SUY RQNG QlJ'(Rll)

Kh6ng gian cac ham suy r(>ng (jj)'(Rll)ra dai la m(>t ta't y€u, tuy v~y, s1f

thi€u v~ng phep loan quail tn;mg nhu phep nhan dff lam cho qua trlnh khao sat

@'(Rll) tra Den truu tuQng va h,;m ch€ v€ phuong phap

Trong phgn nay cua lu~n van se chung minh m(>t sO' m~nh d€ v€ s1f

nhilng qjY(Rll)vao y[Rll] da duQc phat bi€u trong [2] vdi hy v<;mgbudc dgu t~o

co sa ma r(>ng phuong phap khao sat (jj)'(Rll)nha c6ng ClJphong phil cua giai

tich cd di€n D6ng thai thli' ap dlJng cac k€t qua do d€ giai quy€t m(>tvai va'n

d€ ClJth€, don gian trong kh6ng gian (jj)'(Rll).

I vAl KIEN THUC Md DAD:

.Djnh nghla 3.1:

Gia sli'T E (jj)'(Rll),ta d~t

RT(cp,x) =<T(t), cp(t - x», cp E AI, X E Rll

M~nh di 3.2:

i) lim.f<p£(t-X)CD(X)dx = (O(t), Vcp E ,d1, CDE (jj)(Rll)

£ >0

R"

ii) Vdi T E QlY(Rll),ta co

<T, CD>= lim+ fRT(CPE' x)CD(x)dx, Vcp E ,AI, CDE (jj)(Rll)

Chang minh:

i) La'y day Ek -+ 0+ khi k -+ + 00 , ta co

(

t- x

JCD(x)dx

= hm+ f<p (u)W(t-uE:k)du (d6i bie'n u =I-X)

suy fa: £ ->0+hm f<P£(t-x)w(x)dx-w(t),

= hm f<p(u)w(t - UE:k)du - f<p(u)w(t)du

£k->O+

=6'~~+ f<pCu)[ w(t - UE: k ) - w(t) ]du

SUp p'p

do co, cp E @(Rll) Den I cp(u)I, Ico(t - UEk)I, Ico(t)I < c d6ng thai co, cp lien tlJc va suppcp compact Suy ra

hm <p(u)[W(t-UE:k)-W(t)]= 0 va I cp(u) [co(t - UEk)- co(t)] I < c2

£k ->0+

A.p dlJng dinh ly hQi W bi ch~n cho day ham

ta duQc

fk(u) =cp(u)[co(t - UEk) - co(t)]

hm f<p(u) [w(t-UE:k)-w(t)]du = 0

£k ->0+

sup p'p

.

im f<p6'(t- x) w(x)dx = wet)

6'->0+

R"

ii) duQc suy ra tu i, tu dinh ngh'ia tich phan, tu Hnh tuye'n tfnh lien Wc cua

T va cac ham cp, co E @(Rll)

Vi dl,l3.3:

+ Vdi 8 la ham Didc: <8, co> =coCO),V co E qj)(Rll) ta co

Rb(cp, x) = cp(-x), Vcr E Jib Vx E Rll

{

I n€u x > 0 + Vdi ham Heaviside: H: R ~ R, H(x) = /

0 neu x < 0

Taco RH(cp,x)= fcp(t-x)dt

0

Cho R( cp,x) E dl1Ril] Khi do voi m6i co E @[Ril] luon t6n t(;lis6 hI nhien

N sao cho \:f cpE ~dN ta luon co:

}~~ fR(CPc,x)w(x)dx =0, \:fcp E ~dN R"

Chang minh:

Ta co K =suppco la t~p hQp compact Do R( cp,x) E dt:[Ril] lien co s6 tlf

nhien N} va day Y E r sao cho voi m6i cpE ~q (q ~ N) d€u t6n t(;li2 s6 dl1dng C} va 11thoa:

&1

do day y(q) tie'n toi +00lien co th~ chQn q du IOn (q ;:::N ;:::N}) d~ y(q) > Nt,

\:fq;:::N Khi do I R(cp£,x) I ::; C} Ey(q)-Nl ~ 0 khi E ~ 0+,

ChQn day Ek ~ 0+ sao cho Ek E (0,11).

Voi cpE ~dq (q ~ N) va do co(x) bi ch~n lien ta co:

I fR(cp"k,X)W(x)dxI = I fR(cp£k,x)co(x)dx I ::; IIR(CP£k,x) Ilco(x)ldx

K

V~y" 70+lim fR(cp",x)w(x)dx = 0 dung voi mQi cpE~ q

R"

Mfnh d€ 3.5: (Da phat bi~u va chung minh trong [8]).

Voi co E Ql)(R)ta co

fw(x)du = 0<=>::3 co* E @(R) thoa (co*)' = co.

R

II VE BAO HAM THUC QlJ'(Rll) c y[Rll]

(Ham suy rQng QlJ'(Rll)cling la ham Colombeau)

Trong ph~n nay chung Wi chung minh cac mt%nhd€ da: du'Qc phat bi€u

trong [2] v€ va'n d€ nhung kh6ng gian qj)'(Rll) vao kh6ng gian Colombeau

q[Rll].

-Mfnh di 3.6:

Vdi T E qj)'(Rll), ta co RT(cP,x) E ~M[Rll]

Chang minh:

V di t?P compact K c Rll va da chi so a GQi ~ la qua c~u dong co Him t<;li

goc 0 va chua t?P compact K khi 8 dli nho (0 < 8 < 11< 1) ta se co t - X E

8 Stipp cpkeo theo t E B (dung vdi mQi x thuQc K)

Sur fa f(l) =<pea)(1~ x) E fl)(B)

Do T E f!l)'(Rll)va B cling la t?P compact lien co so tl! nhien k va so

du'ong C1 d€

<T,<p(aJ(t:X»IClllf(t)IIk =c11Icp(aJllk~Cl'S\ ,cz ChQn N = k + n + I a I , khi do vdi m6i cPE J'£N ta co

I Du RT«pc, x) I = loa <T(t), <p, (t - xJ> I = En.!j a I < T(t), <p(u)( t ~ x) >

~ sn+llal+kC1CZ= s~ ' dung \Ix E K, \18 E (0, 11)

V?y RT(cp, x) E ~M[Rll].

.Mrnh d€ 3.7:

f!j)'(Rll)la khang gian tuye'n tinh con cua y[Rll] nhd phep nhung

Trong do

j: @'(Rll) + y[Rll]

jeT) =RT( cP,x) + ,h[Rll]

Chung minh:

+ j la anh x~ tuye'n tinh dU<;1csuy ra tu dinh nghla cua RT(cP,x) va tinh tuye'n tinh cua T

+ Bay gid chung minh j la don anh tuc la tu gia thie't jeT) =0 trong

y[Rll] cgn suy ra T =0 trong f!j)1(Rll)

Th~t v~y, do jeT) =0 lien RT(cp, x) E u11Rll]

Voi co E f!j)(Rll),t6n t~i s6 tlf nhien N sao cho

!~~ fRT(q>",x)m(x)dx= 0 , vcp E u4q (q ~ N) (m~nh d~ 3.4)

R"

ma <T, co>=,, >0+hm fRT(q>c'x)m(x)dx (m~nh d~ 3.2)

R"

~ <T, co> =0, vco E Qj)(Rll).

V~y T =0 trong f!j)'(Rll).

.Mrnh d€ 3.8:

Cho T E f!j)'(Rll)va R(cp, x) la mQt d~i di~n cua jeT) (j la phep nhung trong m~nh d~ 3.7) khi do voi m6i co E @(Rll) luan t6n t~i s6 tlf nhien N sao

cho VcpE YiN ta co

1

.

im fR(q>",x)m(x)dx =<T, co>

,, >0+

R"

Chung minh m~nh d~ nay dU<;1csuy ra tu 3.4 va 3.2

M?nh di 3.9:

Voi f(x) E COO(Rll),ta co

Rl(CP, x) = ff(t)<p (t-x)dt va R2(CP,x) = f(x)

R"

la hai d'.li di~n cua mQt ph~n tii' trong q[Rll].

Chang minh:

C~n chung minh Rl(CP, x) - R2(CP,x) E u11Rll]

(vlly do ky hi~u, lien chi chung minh tru'ong hQp n = 1).

Th~t v~y: voi t~p hQpcompact K c R va da chi s6 a

+ Khi a =0:

Ta co I Do- R1(CPE,x) I = I Rl(CPE,x) I = I ~ff(t)<P(

t-x

) dt

I

C:R c:

= I ff(x+uC:)<p(u)du I

R

(d01 len u;>' b'A" =-t-x => t =x + UE).

E

=>

I Do- R1(CPE,x) - Do- R2(CPE'x) I = I ff(x+uc:)<p(u)du- ff(x)<p(~)du I

= I fcP(u )[ f (x + U6') - f (x)] du I

GQi B la qua c~u dong Him t'.li g6c 0 chua K

Voi E du be (0 < E < 11< 1) ta se co U E Stipp cPkeo theo t E B

ChQn N =1, y(q) =q + 2,

Voi cP E x1q\ x1q+l, khai tri6n Taylor ham f(x) t'.li x Wi cap q ta du'QC

f(x + U E) - f(x) =LUJEJf(x).-;-+uq+lEq+lf (8)

trong d6 8 n~m giua x va x + liE,voi x va t thuQCB lien 8 cling thuQc B sur ra

I cp(u)uq+lf (8) (q+ 1)' I ::;CI (CI phl;l thuQc cp), sii'dl;lng R fuP<p(u)du=0 (1 ::;~

::;q) ta du'Qc

If<p(x)(f(x + u&)- f(x)du I = If<p(u)& q+l uq+lf (q+l)(8). 1 I

< C q+lC =C y(q) (c la do do cua Sti pp m).

E

+ Khi a ~ 1: lam tu'dng tlf nhu' tren, thay VI v~n dl;lng cong thuc Taylor

(a)

voi ham f(x), bay giOla ham g(x) = f (x) ta cling du'Qcdi~u c~n chung minh V~y [RI - R2]E JV[R] Do d6 RI(CP,x) va R2(cp,x) la hai d(;lidi~n cua mQtph~n tii'trong y[R].

M~nh dl 3.10:

qj)'(Rll)la khong gian tuye'n tinh con thlfc slf cua y[Rll] (hi~u theo nghla

nhung).

Chang minh:

Ta tha'y 82 thuQc y[Rll] nhu'ng khong thuQc qj)'[Rll].

Th~t v~y: Gia sa 82 = T E 9lJ'(Rll)khi d6 (j trong y[Rll], T se c6 hai d(;li

di~n la RI(CP,x) = cp2(-x) va R2(CP,x) =RT(cp, x) => [RI - R2]E JV[iRll] lien

voi m6i co E qj)(Rll)d~u t6n t(;lis6 tlf nhien N sao cho VcPE J1q (q ~ N) ta d~u

c6

!~~ feR! -R2)«PE,x)OJ(x)dx = 0

R"

(m~nh d~ 3.4)

ma E~Olil11fR2 «p",x)w(x) dx = <T, co>, Vcp E J11

R"

(m~nh d~ 3.2)

sur ra !~~ fRI «pI"x)OJ(x)dx =<T, co>, Vcp E J1q (q ~ N)

R"

m~t khac: ,,~o+lim fRI «pI"x )OJ(x)dx = lil11 1'->0f<p~(-x) OJ(x) dx

R" R"

(d01A!' blen U''" = - -x )

E

= lim(-lY 2- fcp2 (u)OJ(-uE:)du, K = Stipp cpla t?P compact

17 ->0+ E:n

K

Bay gio chQn day Ek ~ 0+ va co E QlJ(Rn) sao cho coCO)*- 0, sa d\lng tinh

compact cua K, tinh bi ch~n cua cp,cova v?n d\lng dinh ly hQi W bi ch~n cho

day ham fk (u) = cp2(u) co(-UEk) ta duQc

lim+ fcp2(u)co(-UEk)du = fcp2(u)CO(0)du = coCO)fcp2(u)du *-0

Ek~O KKK

suy ra Jim (_1)n+ ~n fcp(U)co(-UEk)du = 00

do do lim fRI(CPe'x) OJ(x) dx =00

17 ->0+

R"

di~u nay mall thu~n vdi lim fRJCPe, x) OJ(x)dx = lim fR2(CPe,x)OJ(x)dx

17 ->0+ 17 ->0+

R" R"

= <T, co>

(da co duQctu ph~n tren cua chung minh)

V?y 82 ~ QlJ'(Rn).

.M~nh di 3.11:

V di T E QlJ'[Rn]ta co:

i) D~RT(CP,X) = R a (cp,x)D T

ii) D<Xj(T) =j(D<XT)

Chang minh:

i) D~RT(CP,x)=D~ <T(t),cp(t-x) > = <T(t),D~cp(t-x»

=<T(t), (-1) Ia I <p(a)(t - x» =(-1) I a I < T(t), <p(a)(t- x»

RDaT «p,x) = < DaT(t), <pet- x) > = (_1)1 a I < T(t), D~<p(t - x) >

=(_1)lal <T(t), <p(a>Ct- x»

nhu v~y ta co i)

con ii) ducjc suy ra tn!c ti€p tu i)

III MOT SO KET QUA KHAC:

Cae phuong trlnh Y' =0, Y' =T trong [!J)'(R)dff ducjc giai trong [8] b~ng

phuong phap thac tri~n toan ta tuye'n tinh Trong ph~n nay chung Wi lieU each giai khac dtja tren co sa cae ke't qua dff d~t ducjc khi khao sat cae ham Colombeau.

* Gia sa Y E [!J)'(R)la nghi<$mcua phuong trinh, tu m<$nhd~ 3.11

=> R'y«p, x) =RY'«p, x) =Ro«p, x) =0, V<p E xiI, X E R.

Ta tha'y: Khi c6 dinh <PI E J'iI, voi m6i co E [!J)(R),t6n t~i duy nha't

COoE [!J)(R) thoa co(x) =<PI(x) fm(~)d~+wo(x)va fmo(~)d~ =o.

Ta co <Y, CO>=Ii-+O+!imfRy(cpli,x)m(x)dx, <p E J'il (M<$nh d~ 3.2)

R

=> <Y, co> = !~~

[R fRy (cp0"X)CPI(x)dx fm(C;)dC; + fRy (cpIi' x)mo (X)dX R R ]

M~t khac Ii-+O+!imfRy (CPIi'X)cpJ(x)dx =< Y,cpJ >= a E(['

R

lim fRy (CPIi,x)mo (x)dx = lim

~

l

+oo

- JR" «j>pX)[WoC';-)d';-dx] ~ 0

Suy fa <Y, co>=- fam(x)dx

R

NguQc l:;ti, khong ma'y kho khan khi ki~m l:;ti dng mQi ham

<Y, co> = - fam(x)dx (a la hang s6 thuQc <C)d€u nghit%mPT 3.12.

R

V~y nghit%mcua phuong trlnh 3.121a ham hang theo nghia phan b6.

.Phztdng trinh 3.13: Y' =T tfong f!/j'(R)

* Gia sa Y E f!/j'(R)la nghit%mcua phuong trlnh

=> R'y(cp, x) =Ryo(cp, x) =RT(cp, x), Vcr E J<£1,x ER.

c6 dinh cp E ,;:21, voi m6i co E f!/j(R), t6n t:;ti duy nha't COoE f!/j(R).

Thoa co(x)=CPl(X)fm(()d( + coo(X)va fmo(x)dx= O.

Ta co <Y, co>=,,~o+hm fRy«p",x)m (x)dx

R

= !~~

[ R fRy «p",x)<p}(x)dx fm(()d( R + fRy «p",x)mo(X)dx R ]

Ma li~ fRy «P",X)<PI (x)dx = < Y,<p} >= a EC

C~O

R

hm fRy«p",x)mo (x)dx = lim

[

- JR', (q>"X)}j)o(.;)d~ch]

x

= 0- hm fRT«P",X) fmo(()d(dx

,,~o+

x

= - < T, fcoo(~)d~ >

(do MD 3.5 lien JO)o(~)d~E QiJ(R

-00

suy fa:

x

<Y, 0» = fam(x)dx - < T, fmo(~)d~ >

* NguQc l~i, ta co th€ ki€m duQc ding mQi ham

x

< Y,m >=- < T, fmo(~)d~ > + fam(x)dx d~u thuQc QiJ'(R)va thoa PT 3.13

R

(tfong do fmo(x)dx=O, m(t)=<1'I(t).fm(x)dx+mo(t), CP1c6 dinh thuQc J41,

con a la h~ng s6 phuc thuQc <C

x

V~y nghi~m cua phudng trlnh 3.13 la ham suy fQng < T,- fmo(~)d~ >

-00

cQng vdi ham h~ng (theo nghia phan b6}