Luận văn Hàm suy rộng colombeau

CHUaNG I MOT s6 KIEN THUC CHUAN BI. . VIE HAM COLOMBEAU I. T~P'H<)PtYiq .Dinh nghla1.1: Voi q =1,2,... ta gQiudqla t~phQpcachamcpE @(Rll)thoacacdi~u ki~nsail: f <pet)dt = 1 R" Ita <p(t)dt =0 R" vail:::; lal:::; q .H~qua1.2: n Vai ~(A)=(27r)-Zfe-iAx<p(x)dx, (A E Rll) la anhFouriercuacp(x)thuQcudq, R" . taco: n 1\ -- i) cp(O)=(211:)2 ii) Da ~(O)=0, 1:::; I a I :::;q Changminh: n n i) ~(O)=(27r)-z f <p(x)dx=(27r)-Z R" --------- ii) Theo [1]taco «iA)~Da;)(A) =D~((-ix)acp(X))(A) 5 ChQn~=(0,0, ...,0) taco A ~ =>Da <peA)= (-iX)lh <p(x)(X) A ~ na /'" a .. ~ -- f=>D <p(0)= (-IX) <p(X)(O)= (2JZ")2 (-ixt<p(x)dx=0 R" -Bjnh Ii 1.3: i) ,;4q=I-0 voi mQiq = 1,2,... ii) ,;41=:;,;42=:;... =:;,;4q=:;..;4q+1=:;... CX) iii) n ,;4q=0 q=l Chang minh.. (Vlly do ky hi~ulien chI chungminh tHronghQpn = 1) i) GQi <pet)E YJ(R) thoa f<p(t)dt= 1(trong rdJ(R)hi€n nhienco hamnhu R v~y). Bay giOta xet day ham sail day: <P1(t)=<pet)+al<P'(t) () () (q+1) <Pq+1t = <Pqt + aq+1<P (t) Ta luon tIm duQcaq thichhQpthuQcC d€ <PqE ..dq Th~tv~y: Voi q = 1ta co: f<pj(t)dt= f<p(t)dt+a}f<p'(t)dt R R R = 1 + al.O = 1 6 Jt<pj(t)dt=Jt<p(t)dt+ajJt<p'(t)dt R R R = Jt<p(t)dt - a] J<p(t)dt R R = Jt<p(t)dt-a\ R =>ChQn a] =Jt<p(t)dt ta se COCPlE --Jd1 R Voi qtuyY thuQcN: quyn~ptheoq tasetimduQcaq E C d€ CPqE ,Jdq ii) DuQcsuyratn!cti6ptudinhnghla. iii) Giasa (jJ n .Jdq* 0 q=1 =>t6n t~icPE ~q, q =1,2,... Theo[1],voidinhly Paley-Wienertaco: A <p(~),~E <C lahamnguyenthoa:yoi mQiN >0,t6nt~iCN>0,yoi mQi~E C tad~uco: I I\ (.\:) I <CNexp(Rllm~l)(1) U 1 , ?;:: " ?,.:' d" b/ k/ hR cP'? - N ' R a qua cali tam (j goc tQa Q, an III , (1+I ~I) /\ chuaStippcpoKhai tri€n Mclaurinhamnguyencp(~)t~ig6ctQadQva sad\lng 1\ Do. cp(O) =0voimQidachIs6a ( I a I ~ 1)tathuduQc: 1\ 1\ j cp(~)= cP(0) = (2;r)-2:, voi mQi ~ E <C. 7 Khi lay~E R vachoI ~I c1ulOn,taseco c1U<;1cmallthu~nvoi batc1~ng thuc(1). V~y r:JJ n ~q =0. q=l II. s6 PHUC SUYRONG .Djnh nghla1.4: Voi cp(x)E QlJ(Rn),tac1i;itCPE(X)=~cp ( ~ J , x ERn, E>O. En E Tli c1inhnghla1.1,1.4vadungphudngphap c16ibie'ns6tad~dangchung minhc1lfdch~quasailc1ay. .Hf qua 1.5: Ne'ucPE ,>:iqthl CPE ~q . . Djnhnghla1.6: i) $'0 la t~phQpcachamR(cp)tli ,->:ilVaGC. ii) $'Mla t~ph<;1pcachamR trong$'0 ,thoac1i~uki~n:t6nt'.lis6tlfnhien N E N (N phl;l thuQc R) saG cho voi m6i cPE ~N c1~utlm c1U<;1c2 s6 dudngC va 11c1€ I R( CPE)I ~ ~, '\IEE (0,11).E iii) J lat~ph<;1pcachams6R(cp)trong$'0' thoac1i~uki~n: T6n t'.lis6tlf nhienN E N va dayy E r (N, yphl;lthuQcR) saGchovoi m6i cPE ,>:iq(q ~N) c1~utlm c1u<;1c2 s6 dudngC va 11c1€ I R(CPf;) I ~ CEy(q)-N, '\IE E (0, 11). .Djnh ly1.7: i) $'Mla mQtvanhvoi phepcQngvanhananhX'.l. ii)J lamQtIc1eancua$'M (Chung minh c1inhly nay xin trlnh bay (j ph~nphl;lchudng) 8 .fJjnh nghla1.8: T~phQpcac s6 phuc suyr<)ngla vanh thudng ~M/J, kyhi~ula C. .M~nhdi 1.9:C la vanhconcuaC nhophepnhungj: C ~ Ctrongdovdi ZE<Ctacoj(z) =R(CPIJ+J ddoR(CPE)=Z, Vcp E~Y11,V8>0. Changminh: +j la anh x:;tVI: chc,>llN =1=> V cP=~Y11taco: I R( CPE)I =I Z I::;; ~,vdiC =I z I , 11=1,8 E (0, 11)8 => R(cp) E ~M =>j(z) E C. +j lad6ngc§uvanh:hi~nnhien +j la donanhVI: n€uj(z) =0 E C. =>R(cp)EJ =>T6n t:;tis6N E N va dayy E r d~vdi cPE ~~q(q :2:N) d~uco hai s6 du'ongC, 11saocho I Z I =I R(CPE) I ::;; C.8;q) =C 8y(q)-N~ 0khichQnqduldnsaocho y(q)>N8 vacho8~ 0+. Suyraz=0 E C Tli m~nhd~trentacoz EJ q Z=0 (duQchi~utheonghlanhung). ID.HAMSUYRONGCOLOMBEAU Ky hi~u:~O[Rll]la t~phQpcac anhx:;tR tli ~l x Rll vao <Cma khi c6 dinhcPtaduQcR(cP,x) khclvi mQic§ptheox. 9 .EJjnh nghia1.10: T~p h<jp$'M[Rll]g6m cac ham R(cp,x) trong $'0[Rll] thoa vdi mQI t~pcompactK c Rll va dachi s6a d~ut6ntqi s6h! nhienN EN saochovdi m6icpE JiN d~uHmdu'<jc2 s6du'dngC va 11d~ I oCtR( CPf;,x) I ~ ~ dungE Vx E K, VE E (0,11)(trongdooCtRia dqohamdip I a I cuaR theobienx). .EJjnhnghia1.11: T~ph<jp.A1Rll]g6mcachamR(cp,x) cua $'0[Rll] thoa vdi mQi t~p compactK c Rll va dachi s6a d~ut6ntqi s6tt!nhienN va dayY E r saocho vdim6icPE Jig (q~N) d~uHmdu'<jc2 s6du'dngC va 11d~: C Er(g) I oCt R(CPf;,x) I ~ . N dung Vx E K, VE E (0,11). E .EJjnh Ij 1.12: i) $'M[Rll]la mQtvanhvdiphepcQngvanhananhXq. ii) QJf/[Rll] la ideancua $'M[Rll] (xin trlnhbayb~ngchungminhdinhly nayd phgnphl;!chu'dng) .EJjnhIj 1.13: T~ph<jpcachamsuyrQngColombeaula vanhthu'dng$'M[Rll]/./11Rll]. Ky hi~ula y[Rll]. Tudinhnghla1.13tacocacdi~usailday: + M6i hamsuyrQngColombeaulamQtlOptu'dngdu'dngcodqng: G =R(cp,x) +./11Rll],R(cp,x) E $'M[Rll]la mQtdqidi~ncuaG. +PhepcQngvaphepnhanhaihamsuyrQngColombaeudu'<jcth\fchi~n nhu'sail: 10 G1 + G2 =Rl(cp,x) +R2(cp,x) +J1;[Rll] G1.G2=Rl(CP,x).R2(cp,x) +J1;[Rll] Trongd6R1(cp,x) la d(;lidi~ncuaG1,R2(cp,x) la d(;lidi~ncuaG2. IV. GIA.TRJ CUA HAM SUY RONG COLOMBEAU T~I Xo E Rll .FJjnh nghia 1.14.- Gia tri cua ham suy rQngColombeau G t(;liXo E Rll la so phuc suy rQng R(cp,xo)+:J E C. Trongd6R(cp,x) la mQtd(;lidi~ncuaG trong~M[Rll]. D~dinhnghla1.14hejpl~c~nphaichungminhhai di~usailday: i) R(cp,xo)E ~M ii) Dinh nghla1.14khongphl;!thuQcvaod(;lidi~n. Th~tv~y: i) La'yt~pcompactK c Rll chuaXo,dachI soa, I a I =O. Do R(cp,x) E ~M[Rll] t5n t(;liso tlf nhien N d~vdi m6i cpE udN,d~utIm du'ejc2 so du'ongC va 11saocho: IDa R(CPE'x) I ~ ~, Vx E K, VE E (0,11) E => I R( CPE,xo) I ~ ~ E => R( cp,xo) E ~M ii) Gia saR1,R21ahaid(;lidi~ncuaG. =>(Rl - R2)E J1;[Rll] =>Vdi t~pcompactK chuaXodasochIsoa, I a I =O.T5nt(;lisotlf nhienN va day soY E r saocho vdi m6i cpE .Yiq(q ~N) d~utImdu'ejc2 so du'ongC va 11d~: C Ey(g) IDa (R1 - R2)(CPE,x) I ~ . N ' \:Ix E K, \:IE E (0,11). E C.Ey(q) => I (R1 - R2)(CPE, Xo) I ~ ~ E => I (R1 - R2)(CPE'xo) I EJ => R1(cp,xo) va R2(cP,xo) la hai d~idi~ncua mQts6 phucsuy rQngtrong C. , ?, A v. D~OHAM CUA HAM SUY RQNG COLOMBEAU .fJjnh nghla1.15: Da(G) =DaR(cP,x) +JI1Rll], trongdo G E y[Rll] co d~idi~nla R(cP,x), DaR(cP,x) la d~ohamdtp a cuaR(cP,x) theobienx, cona la daChIs6tuyy. . £)6dinhnghIa1.15hQpl~tac§nchungminhhaidi€u sail: i) DaR(cp,x) E $'M[Rll] ii) Da(G) khongphl;lthuQcvaod~idi~n. Th~tv~y: i) Voi t~pcompactuyyK vadaChIs6~=> ~+a clingla daChIs6. Do R(cp,x) E $'M[Rll] lien co s6 tl! nhien N sao cho voi m6i cpE ~JiiNd€u tlm du'Qc2 s6 du'c5ngC va 11d6 I a+13 I C D R(CPE'x) ~ N' \:Ix E K, \:IEE (0,11).E I 13 a I C => D (D R(CPE'X) ~ N' E => Da R( cP,X) E $'M[Rll] 12 ii) Giii suR},R2la haid(;lidi~ncuaG =>R =Rl - R2 E ut[Rll].Voi t?P compactK chuaXo,dachi sf)~(a va~ladachisf)liena +~clingla dachi sf)) Do R E .A1Rll]lienvoi mQit?P compactK c Rll va dachi sf)~+a cosf) tl!nhienN vadaysf)Y E r saochovoim6i cpE '~q(q~N) d~utimduQc2 sf)dudngC va11d~: C Ey(q) I Da+PR(q\;,x) I ~ . N ' '\Ix E K, '\IE E (0, 11). E => I D~(D<XR(cpc;,x) I =I D~(D<X(R1 - R2)(CPc;,x) I C.Ey(q) =I D~(D<XRl - D<XR2(cpc;,x)) I ~ ~ E =>(D<XRl - D<XR2)(CP,x) E ut[Rll]. . Tli dinhnghla1.15vachungminhtrentacom~nhd~sail: .Mfnh di 1.16: i) M6i hamColombeaud~ucod(;lohammQica'p. ii) D(;lohamcuaColombeauclingla hamColombeau .Mfnh di 1.17: CongthucLeibnitzv~ncondungvoid(;lohamcuahamColombeau D<X(GIG2)= ICt.Da-fJGJ.DfJG2. O~fJ~a Th?t V?y:Voi R1(cp,x) la d(;lidi~ncuaGl, R2(cp,x) la d(;lidi~ncuaG2. D<X(GIG2) =D<X(RIR2)+ut[Rll] = ICt.Da-fJ R1.DfJR2 +ut[Rll] O~fJ~a = ICt.Da-fJGJ.DfJG2 O~fJ~a 13 VI. PHEP NHAN, PHEP CONG SO PHUC SUY RONG (HO~CSO PHUC)VOl HAM COLOMBEAU. .M~nhdi 1.18: Vanhs6phucsuyrQngCduQcnhungvaovanhcachamColombeau y[Rn]nhoghepnhung j: C ~ y[Rn] voi j(c)=keep)+J1'[Rn] trongdoc E C cod(;lidi<%nk(ep). Changminh: Truoche'tta co nh~nxetdng: Tli cacdinhnghla1.6,1.10va 1.11d~ dangcoduQc acdi~usailday: e N€u R(ep)E $'Mthl R(ep)E $'M[Rn] eNe'uR(ep) EJthlR(ep) E J1'[Rn] eNe'uR(ep)E .h[Rn] vaR(ep)kh6ngphl;!thuQcx thlR(ep)EJ Bay giOtachungminhm<%nhd~1.18theocacbuocsail: +j la anhX(;l,th~tv~y: do keep)E $'M=>keep)E $'M[Rll]=>j(c) E y[Rll] gia sli'k1(ep)la d(;lidi<%nkhac cua k. =>[k1(ep)- k( ep)]E J =>[kl(ep)- keep)]E J1'[Rll]vakeep),k1(ep)E $'M[Rll] =>keep),k1(ep)la haid(;lidi<%ncuamOtph~ntli'trongy[Rll] =>j kh6ngphl;!thuQcvaod(;lidi<%n. V~yj la anhX(;l. +j la d6ngdiu vanh,th~tv~y: 14 Gia sukl(CP)la d(;lidi~ncuacl E C k2(cp)la d(;lidi~ncuac2E C =>k1(cp)+k2(cp)lad(;lidi~ncuacl+c2 E C k1(cp).k2(cp)la d(;lidi~ncuacl.c2 E C Ta co: j(Cl) =k1(cp)+J1;[Rll] j(C2) =k2(cp)+ J1;[Rll] => j(Cl) +j(C2) =k1(cp)+k2(cp)+J1;[Rll] =(k1 + k2)(cp)+ J1;[Rll] =j(Cl + C2) j(Cl).j(C2)=k1(cp).k2(CP)+./V[Rll] =(k 1.k2)(cp)+ J1;[Rll] =j( Cl.C2) =>j lad6ngdiu vanh. +j la ddnanh,th~tv~y: Giasu j(Cl)=j(C2)=>k1(cp)+J1;[Rll] =k2(cp)+J1;[Rll] =>[k1(cp)- k2(cp)]E J1;[Rll] ma [k1(cp)- k2(cp)]khong phl;}thuQcX =>[k1(cp)- k2(cp)]EJ - =>Cl =C2trong C =>j la ddnanh V~yj la ddnca'u. Tli m~nhd€ tren,tacoh~quasailday: -Hf qua1.19: Ne'u: k(cp)la d(;lidi~ncua C E C R(cP,x) la d(;lidi~ncua ham Colombeau G E y[Rll] 15 Thi: j(c) +G =(k(ep)+R(ep,x)) +J1I[Rn] j(c).G =(k(ep).R(ep,x)) +J1I[Rn] . Ta cothtinhungt~ps6phucC vaot~pheJpcachamColombeauy[Rn] nhorichcuahaiphepnhungdanh~cd€n am~nhd~1.9vam~nhd~1.18. . Voi nhungcosatrentadid€n dinhnghlaphepnhan,phepcQngmQts6 phucsuyrQngc (ho~cs6phucZ E C) yoi hamColombeaunhu'sail: .Dinh nghia1.20: Giasa c E C cod~idi~nlak(ep) G E y[Rn]cod~idi~nla R(ep,x) Zla s6phucthuQcC Ta gQi:c.G =k(ep).R(ep,x) +,/11Rn] c +G =keep)+R(cp,x) +J1I[Rn] Z.G =z.R(ep,x) +J1I[Rn] Z+G =z +R(ep,x) +J1I[Rn] . Tli dinhnghla1.20tatha'yvanhy[Rn]la mQtkh6nggianvectotren tru'ongs6phucC. 16