Hàm suy rộng Colombeau

€ giai quy€t m(>tvai va'n d€ ClJth€, dongiantrongkh6nggian(jj)'(Rll). I. vAl KIEN THUC Md DAD: .Djnh nghla3.1: Gia sli'T E (jj)'(Rll),tad~t RT(cp,x) =<T(t), cp(t- x», cpE .AI, X E Rll. .M~nhdi 3.2: i) lim.f0 R" ii) Vdi T E QlY(Rll),taco =lim+ fRT(CPE' x)CD(x)dx, Vcp E ,AI, CDE (jj)(Rll) E~O Rll Chang minh: i) La'ydayEk-+ 0+khi k -+ +00 , taco lim f<p& (t - x)m(x)dx=lim f~<p( t - x J CD(x)dx & -->0+ k & ->0+ C' C' k R" k R" (J k (J k 24 =hm+f<p(u)W(t-uE:k)du (d6ibie'nu =I-X) 6',->0R" E:k suy fa: hm f0+ , k R" = hm fO+ R" Rn =6'~~+ f<pCu)[ w(t - UE:k )- w(t)]du SUp p'p do co,cpE @(Rll) Den I cp(u)I, I co(t- UEk)I, I co(t)I < c d6ng thai co,cplien tlJcva suppcpcompact.Suyra hm<p(u)[W(t-UE:k)-W(t)]=0 vaI cp(u)[co(t- UEk)- co(t)]I <c2 £k ->0+ A.pdlJngdinhly hQiWbi ch~nchodayham taduQc fk(u)=cp(u)[co(t- UEk)- co(t)] hm f0+ supp'p V~y 1 . im f0+ R" ii) duQcsuyratui, tudinhngh'iatichphan,tuHnhtuye'ntfnhlienWccua T va cac ham cp,coE @(Rll). .Vi dl,l3.3: +Vdi8lahamDidc: =coCO),V coE qj)(Rll)ta co Rb(cp,x) = cp(-x),Vcr E Jib Vx E Rll { I n€u x >0 +VdihamHeaviside:H: R ~ R, H(x)= / 0 neux <0 25 +00 Taco RH(cp,x)= fcp(t-x)dt 0 .Mfnh d€ 3.4: Cho R(cp,x) E dl1Ril].Khi dovoi m6icoE @[Ril]luont6nt(;lis6hI nhien N saocho \:fcpE ~dNtaluonco: }~~fR(CPc,x)w(x)dx=0, \:fcpE ~dN R" Changminh: Ta coK =suppcola t~phQpcompact.Do R(cp,x) E dt:[Ril]lien co s6tlf nhienN}va dayY E r saochovoi m6icpE ~q (q ~N) d€u t6nt(;li2 s6dl1dng C}va 11thoa: I I C E:y(q) R(cp£, x) ::; 1"N ' \:fx E K, \:fE E (0,11) &1 dodayy(q)tie'ntoi+00liencoth~chQnq duIOn(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 doco(x)bi ch~nlien taco: I fR(cp"k,X)W(x)dxI =I fR(cp£k,x)co(x)dxI ::; II R(CP£k,x) Ilco(x)ldx R" K K ::; fc1&[(q)-NIC ~ 0 khi Ek ~ 0+. K V~ylim fR(cp",x)w(x)dx=0 dungvoi mQicpE ~q"--70+ R" .Mfnh d€ 3.5:(Daphatbi~uvachungminhtrong[8]). Voi coE Ql)(R)taco fw(x)du=0::3co* E @(R) thoa (co*)'=co. R 26 II. VE BAO HAM THUC QlJ'(Rll) c y[Rll] (HamsuyrQngQlJ'(Rll)clingla hamColombeau) Trong ph~nnay chungWi chungminhcac mt%nhd€ da:du'Qcphatbi€u trong [2] v€ va'nd€ nhungkh6nggian qj)'(Rll)vao kh6nggian Colombeau q[Rll]. -Mfnh di 3.6: Vdi T E qj)'(Rll),tacoRT(cP,x) E ~M[Rll] Changminh: Vdi t?P compactK c Rll va da chi so a. GQi ~la quac~udongcoHimt<;li goc0 vachuat?PcompactK. khi8 dli nho (0 < 8 <11< 1) ta se co t- X E 8 Stippcpkeo theo t E B (dungvdi mQix thuQcK). Sur fa f(l)=<pea)(1~x) E fl)(B) Do T E f!l)'(Rll)va B clingla t?Pcompactlienco sotl!nhienk va so du'ongC1 d€ <T,<p(aJ(t:X»IClllf(t)IIk =c11Icp(aJllk~Cl'S\ ,cz ChQnN =k +n + I a I , khi dovdi m6icPE J'£N taco I Du RT«pc,x) I =loaI =En.!j a I ~ sn+llal+kC1CZ=s~ ' dung \Ix E K, \18E (0, 11). V?y RT(cp,x) E ~M[Rll]. 27 .Mrnh d€ 3.7: f!j)'(Rll)akhanggiantuye'ntinhconcuay[Rll]nhdphepnhung Trongdo j: @'(Rll) --+y[Rll] jeT)=RT(cP,x) +,h[Rll] Chungminh: + j la anhx~tuye'ntinhdU<;1csuyra tu dinhnghlacua RT(cP,x) va tinh tuye'ntinhcuaT. + Bay gid chungminhj la donanhtucla tu gia thie'tjeT) =0 trong y[Rll] cgnsuyraT =0 trongf!j)1(Rll). Th~tv~y,dojeT) =0 lien RT(cp,x) E u11Rll]. Voi coE f!j)(Rll),t6nt~is6tlfnhienN saocho !~~fRT(q>",x)m(x)dx=0 , vcpE u4q(q~N) (m~nhd~3.4) R" ma=hm fRT(q>c'x)m(x)dx (m~nhd~3.2),,-->0+ R" ~ =0, vco E Qj)(Rll). V~yT =0 trong f!j)'(Rll). .Mrnh d€ 3.8: Cho T E f!j)'(Rll)va R(cp,x) la mQtd~idi~ncuajeT) (j la phepnhung trongm~nhd~3.7).khi do voi m6icoE @(Rll) luant6nt~is6tlf nhienN sao choVcpE .YiNtaco 1 . im fR(q>",x)m(x)dx =,,-->0+ R" . Chungminhm~nhd~naydU<;1csuyra tu3.4va 3.2 28 .M?nhdi 3.9: Voi f(x) E COO(Rll),taco Rl(CP,x) =ff(t)<p(t-x)dt va R2(CP,x) =f(x) R" la haid'.lidi~ncuamQtph~ntii'trongq[Rll]. Changminh: C~nchungminh Rl(CP,x) - R2(CP,x) E u11Rll] (vlly doky hi~u,lienchi chungminhtru'onghQpn=1). Th~tv~y:voit~phQpcompactK c R vadachis6a +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)duI R (d ;>' b'A" t-x )01 len u =- =>t=x +UE . E => I Do-R1(CPE,x) - Do-R2(CPE'x) I =I ff(x+uc:)<p(u)du-ff(x)<p(~)duI R R =I fcP(u)[ f (x +U6')- f (x)] du I GQiB la quac~udongHimt'.lig6c0 chuaK. Voi Edube (0<E<11< 1)tasecoU E StippcPkeotheot E B. ChQnN =1,y(q)=q +2, Voi cPE .x1q\ .x1q+l,khai tri6n Taylor ham f(x) t'.lix Wi cap q ta du'QC q .. (j) 1 (q+l) 1 f(x +U E) - f(x) =LUJEJf(x).-;-+uq+lEq+lf(8) j=l J! (q+1)! 29 trongd68n~mgiuax vax +liE,voix vatthuQCB lien8clingthuQcB surra (q+l) 1 I cp(u)uq+lf (8) .( 1)' I ::;CI (CI phl;lthuQccp),sii'dl;lng fuP<p(u)du=0 (1 ::;~q+ . R ::;q) ta du'Qc If I If q+l (q+l) 1 I<p(x)(f(x+u&)- f(x)du = <p(u)& uq+lf (8). R R (q+l)! <C q+lC =C y(q) (c ladodocuaStippm).- I.E 2 . N 2. 't' E + Khi a ~1: lam tu'dngtlf nhu'tren,thayVI v~ndl;lngcongthucTaylor (a) voihamf(x),baygiOlahamg(x)=f (x) taclingdu'Qcdi~uc~nchungminh V~y [RI - R2]E JV[R].Dod6RI(CP,x)vaR2(cp,x) lahaid(;lidi~ncua mQtph~ntii'trongy[R]. .M~nhdl 3.10: qj)'(Rll)la khonggiantuye'ntinhconthlfcslfcuay[Rll](hi~utheonghla nhung). Changminh: Ta tha'y82thuQcy[Rll]nhu'ngkhongthuQcqj)'[Rll]. Th~tv~y:Giasa82=T E 9lJ'(Rll)khid6(j trongy[Rll],T sec6haid(;li di~nla RI(CP,x)=cp2(-x)vaR2(CP,x)=RT(cp,x) =>[RI - R2]E JV[iRll] lien voi m6icoE qj)(Rll)d~ut6nt(;lis6tlf nhienN saochoVcPE J1q (q~N) tad~u c6 !~~feR!-R2)«PE,x)OJ(x)dx=0 R" (m~nhd~3.4) ma lil11fR2 «p",x)w(x)dx= ,VcpE J11E~O R" (m~nhd~3.2) sur ra !~~fRI «pI"x)OJ(x)dx =,VcpE J1q (q ~N) R" m~tkhac: limfRI «pI"x)OJ(x)dx = lil11f0 R" R" 30 (d A!' b' '" x )01 len U =- - 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 coE QlJ(Rn)sao cho coCO)*-0, sad\lngtinh compactcuaK, tinhbi ch~ncuacp,cova v?n d\lngdinhly hQiW bi ch~ncho 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~ fcp(U)co(-UEk)du = 00+ n Ek~O Ek K do do lim fRI (CPe'x) OJ(x) dx = 0017--->0+ R" di~unay mall thu~nvdi lim fRJCPe, x) OJ(x)dx = lim fR2(CPe,x)OJ(x)dx17--->0+ 17--->0+ R" R" = (daco duQctuph~ntrencuachungminh). V?y 82 ~ QlJ'(Rn). .M~nhdi 3.11: Vdi T E QlJ'[Rn]ta co: i) D~RT(CP,X)=R a (cp,x)D T ii) D<Xj(T) =j(D<XT) Changminh: i) D~RT(CP,x)=D~=<T(t),D~cp(t-x» 31 =<T(t), (-1) Ia I <p(a)(t- x» =(-1) I a I <T(t), <p(a)(t- x» RDaT«p,x) = =(_1)1aI =(_1)lal Ct-x» nhuv~ytacoi) conii) ducjcsuyratn!cti€p tui) " "" ,,' ? , III. MOT SO KET QUA KHAC: CaephuongtrlnhY' =0, Y' =T trong [!J)'(R)dffducjcgiai trong[8]b~ng phuongphapthactri~ntoantatuye'ntinh.Trongph~nnaychungWilieUeach giai khacdtjatrenco sa caeke'tquadffd~tducjckhi khaosatcaeham Colombeau. .Ph-tldngtrinh 3.12: Y' =0 trong [!J)'(R) * Gia sa Y E [!J)'(R)la nghi<$mcuaphuongtrinh,tum<$nhd~3.11 =>R'y«p, x) =RY'«p,x) =Ro«p,x)=0, V<pE .xiI, X E R. Ta tha'y:Khi c6 dinh <PIE J'iI, voi m6i coE [!J)(R),t6n t~iduy nha't COoE [!J)(R) thoa co(x)=<PI(x) fm(~)d~+wo(x)va fmo(~)d~=o. R R Ta co=!imfRy(cpli,x)m(x)dx,<pE J'il (M<$nhd~ 3.2)Ii-+O+ R =>= !~~[ fRy (cp0"X)CPI(x)dx. fm(C;)dC;+ fRy (cpIi'x)mo(X)dX]R R R M~t khac !imfRy(CPIi'X)cpJ(x)dx ==a E (['Ii-+O+ R lim fRy(CPIi,x)mo(x)dx = lim ~ RY(CPIi'X)' Xfmo(Odc; l +oo 0'-+0+ 0'-+0+R - 00 - JR" «j>pX)[WoC';-)d';-dx] ~0 32 Suyfa =- fam(x)dx R NguQc l:;ti, khong ma'ykho khan khi ki~m l:;ti dng mQi ham =- fam(x)dx(a lahangs6thuQc<C)d€u nghit%mPT 3.12. R V~ynghit%mcuaphuongtrlnh3.121ahamhangtheonghiaphanb6. .Phztdngtrinh 3.13:Y' =T tfong f!/j'(R) * GiasaY E f!/j'(R)langhit%mcuaphuongtrlnh =>R'y(cp,x) =Ryo(cp,x) =RT(cp,x), Vcr E J<£1,x ER. c6 dinh cpE ,;:21,voi m6i coE f!/j(R),t6nt:;tiduy nha'tCOoE f!/j(R). Thoaco(x)=CPl(X)fm(()d( +coo(X)va fmo(x)dx=O. R R Ta co =hmfRy«p",x)m(x)dx,,~o+ R = !~~[ fRy«p",x)<p}(x)dx.fm(()d( +fRy«p",x)mo(X)dx ]R R R Ma li~fRy«P",X)=a ECC~O R hmfRy«p",x)mo(x)dx=lim [ Ry«P,,'X). XfOJo(c;)d('+OI),,~o+ ,,~o+ R -01) 1-01) - JR', (q>"X)}j)o(.;)d~ch] x =0- hmfRT«P",X)fmo(()d(dx,,~o+ R -01) x =- -00 33 x(doMD 3.5lien JO)o(~)d~E QiJ(R -00 suyfa: x R -00 * NguQcl~i,tacoth€ ki€m duQcdingmQiham x =- +fam(x)dxd~uthuQcQiJ'(R)vathoaPT 3.13 R (tfong do fmo(x)dx=O,m(t)=<1'I(t).fm(x)dx+mo(t),CP1c6 dinh thuQc J41, R R cona la h~ngs6phucthuQc<C. x V~ynghi~mcuaphudngtrlnh3.13la hamsuyfQng -00 cQngvdi hamh~ng(theonghiaphanb6} 34 ._.