Luận án Về một số bài toán ngược trong phương pháp trọng lực

Chztcmg2 :Bai Torin(]!1-y,chyChoPhuangTrinh Laplace . Chztdng 2. BAI ToAN CAUCHY CHO PHUONG TRINH ' LAPLACE. I. Gidi thi~u Bai toanCauchyrho phucmgtrinh LaplacedaduqcHadamarddclutien chl.rala bai tminkhongchinh.Trongkhi dobai toanDirichlet rho phucrng trinh Laplacela bai tmin d~tdung.Tuy nhien, khoa hcva ky thu~tl~i yell do, trong mQt 86 trucrnghqp, phai xet bai toan Cauchy rho lo~i phucrng I trlnhnoi tren.Thf d1;1,ngucrita mu6nxacdinh di thucrngtrngIt!C(mQtd~i ltiqngthoaphucrngtrinh Laplace)a bell tren m~td~t,do cacdi v~ta bell I dti6im~td~tgay nell, tit cac'86 li~u do d~cth1!chi~n tren mQttuye'nho~c ffiQtvung.Hai mohinh toan quellthuQcla bai toanDirichletho~cNewnann chophucrngtrinh Laplacetren Ill/a m~tphiing tren hay Ill/a khong gian tren dtiqcdungd~tlnh toan.Tren 1:9'thuye't,cac mo hinh toan tren dbi hoi cac dO'ki~ndo tren mQtducrngthiingdai vo h~nho~cmQtm~tph~ngrQngvo h~n,dayla di~ukhongkhii thi. V6i mohinh la bai toanCauchyrho phucrng trinh Laplace tren Ill/a m~tphiing tren ho~cmla khong giaIi tren ngucrita chi cdn dii'ki~n do tren mQttuye'nco dQdai hii'uh~n,ho~cmQtph~nm~t. phiingcodi~ntlch hUllh~n;di~unayco duqcla docacdinh 1:9'v~duynh~t Ilghi~mcuabai toanCauchyrho phucrngtrinh Laplace[23].Tuy nhien,nhu daneutren,bai toannayla khongchinh,nghIala, nghi~mkhongloonloon, ,", . j, " 14 Chuang2 : Ba,i.f()an,_p(J,l!cl'1,~~hoP ~angJ,}:Lnf/,_~(J,Rtace tant~i,va ne'utan t~inghi~m,thi nghi~mkhongtuy thuQclien t1,lCvancac di~uki~nbien. TrongHnh toanth1!Cd1,lng,cacbai toankhongchinh cc1nphiii duqc chinhboa.Nghi~mchinhhoala mQtnghi~mxa'pxi 6n dinh. Va'nd~chinh hoabai toanCauchy chophuangtrlnh Laplacetrongtru<Jnghqpt6ngquat da:duqcLattesS. va Lions J.L. xet trong[37].Cactacgia nay dungphuang phapquasi-reversibilitedg chinh hoa bai toan, nht1ngkhong dua ra u6c htqngv~sai s6. ChungWi dungphtfangphapcWnhhoa bai toan moment ..~ hUllh~nva u6cluqngduqcsai s6 giO'anghi~mchinhhoava nghi~mchinh xactrongtru<Jnghqpcaedi~uki~nbienbi anhhud'ngbd'isai s6dod~c. Trongchuangnaychungtoi xetbabai toan sauday. 1.Bai tOaDCauchycho phuangtrinh Laplace tren mJ'ami,itphclngtren. . 2. Bai tOaDCauchychophuangtrinh Laplacetren mJ'akhong gian tren. 3. Bai tOaDCauchycho phuangtrinh Laplacetren mQtgiai khong d~u. II. BaitoanCauchychophu'dngtrinhLaplacetren",ram~tph3ngtren. 11.1.Baitoan Bai toanduqcxet la : Tim mQthamu(x,y) di~uhoa tren nO'ami,itphclng tren trongdo D={ (x, y) I - co0 } , lien t1,lCtren 0 = {(x,y)I -co <x co,y ;::: 0 },thoacacdi~uki~Ilu(x,O)=u,,(x) vaau~'y(x,O)==uy(x,O)=Ul(X)v6i -1<x <1.Noi cachkhac,xetbai toaDbien ... 15 Chuan/l..2 : Bai Toan Cauchy Cho_Ph_ucJ1zg_'I'-,"'in"'__~aplace thoa { v2u =0 , trenD , u(x,O)=uo(0) , - 1 <x <1 , uy(x,O)=Ut(x) , -1 <x <1 , (1) (2) (3) trongd6 v2 la tmintil Laplace.Day la bai tminCauchycho phtiCingtrlnh Laplacetren nilam~tphiingtren. 11.2.Thie'tI~pphu'o'ngtrinhtichphanvachTnhh6a Ky hi~u: I =(-1:1), J =R \ I Bai tmin (1), (2), (3) la bai tmin Cauchytren nila m~tphiing tren D. La'ytori gia bien Neumannv(x)=Uy(x;O),X E J la §'nham. Khi d6 ne'uUrn dtiqcv(x), x E J thi Bexac dinh dtiqcu(x,y), (x,y) ED. Nhti v:%ychungta cobai toan Cauchyngtiqc.Chung tOi thie'tl:%pphtiCingtrlnh tfch phan tfnh v. . D~t 1 . N(x,y;~,fJ)=- 27t[In~(x- ~2+(y- fJ)2+In ~(x- ~2+(y +fJ)2] Thi I J 2 2N(x Y.~O)=--ln (x-~ +y, ,-", . 7t I [ -2(y - fJ) 2(y +fJ) l JN"(x,y;~,fJ)=- 47t (x - ~2+(y - fJ)2 + (x - ~2+(y +fJ)2 vadodo N" (x, y; ~,O)=0 HClnnfia ta cfingtha'yding 'hamN thoa phtiCingtrlnh Laplace, v2N =O. XemmQtmi~nR' vai bienS va0'nhtihinh 1 ..to 1~ Chuang2-=-J1qj,TarinCa~cl"LChaPhuangTr'inhLaplace t1 R' s (X'" a ~ 0 L hinh 1 .'" Ap d1;1ngdiingthticGr~enII chocaehamu, N tren mi~nR' vdibienS vaT, chungta duqc. II (uV2N - NV2u)ds=I (u~~ R' s au I oN N -)dl + (u--ov ov 0" N ~~)dl v la phap tuye'nhudngra ngoai mi~nR'. Cho E ~ 0 va L ~ 00, sau ffiQts6 xiip xe'pchungta duqc. u(x,y) = J (N(x,y;l;,,,)u(l;,~)- u(1;,, )N(x,y;l;,,,»)dlv v s. ! I In ~(x_1;)2+y2 w(l;)dl;1t-oo (4) = trongdo W(/;),;, { UI (l;) , l;E I , v(l;) , l;E J ~ .~ Cho y -» 0+, chungta duqc I f 00 u(x~O)=- (In I x - ~l)w(l;)d~1t -00 Vdi x E I , chungta duqc ! I In I x - ~Iv(l;)dl;=Uo(x)_! I In I x -l; IuI(l;)dl;1t 1t J I Ji, 11,., !- Chuang2 : BatToanCauchyChoPhuangTrinh Laplace hay ! f In I x - ~Iv(l;)d~=F(x)1t J (5) trongdo " F(x) =uo(x) -1. f In I x - ~lu1(~)d~1t I (6) Phu<1ngtrinh (5)la phU<1ngtrinh tichphantinh v. Dayla phu<1ngtrinh tich phantuye'ntinh lo~iI. Phu<1ngtrinh nayho~ckhongco nghi~m,ho~c ne'uconghi~mthi covasringhi~m.Nhuv~yva'nd~chiIihhoac~nduqcd~t "" ra.Saud<1ychungtoi chinhhoa(5)b~ngcacbai toaDMomenthooh~n. 11.3.Chlnhh6a. Trong I =(-1,1)la'ymQtday{xJ,I = 1,2, ... . Xet cacphu<1ngtrinh,tich phan. J fnlxi-~v(~d~= <p(xi),iE N, J (7) trongdov thuQcvaoH la khonggianconcuaL2(J)sinhbiHcacham{giheN dQcl~ptuye'ntinh. InlXi - ~I . gj (~)= 2' 1 EN, ~E J I+~ H = Phu<1ngtrinh (7) duqcgQila bai toaDmomenttheoStieltjes.Ky hi~u ( , ) va 11.11Mn luqt la tichvahuangva chu~ntrongL2(J), dinhbi1i L2(J) ={<p'J <p2(~(l+~2)d~< 00 } . J «p,\II) = J<P(~.\jJ(~(l+~2)d~ J ( J II2 II 'I'~ =!'1'2(1+~2)d~ Joi 1~ Chlic1ng2 : Bdi ToanCaUchyChoPhumigTrlnh Laplace . Xet bai tmln V6i m6i Ii E N, tim vn E Hn sao cho (vn,gi) =<Pi,1:$ i :$n (8) trongdo<P= {<Pi},=1,.." n la cacs6th1!Cchotnf6c,Hn la khonggiansinh b?' { } 1 <'<(11 gi, - 1 - n _.- Hn =<gl ..., go) Phucrngtrinh (8)du<JcgQila bai toanmomenthOuh~n,Day la bai toand~t dung,nghIala conghi~mduynha'tvn(CP),tuythuQclien t1,ICvaocp,Th<%tv<%y, ta co n Vn =L ~jgj j=i E Hn (~1,..., ~nE R) la nghi~mcua(6)ne'uva chfne'u n L ~j(gi,gj) j=i = <Pi , i =1,2,...,n. (9) Dayla mQth~phucrngtrlnh tuye'ntinh b<%cn codinhthdc ~=det [ (gi,gj) ] \ :<;;i.j :<;;n ~chfnhla dinhthdcGramcuah~gt , , gn. Vi gt ,...,gn dQcl<%ptuye'ntinh nen~:j;0,v<%y(9)la h~Cramerva do do xac dinh (~l, ..., ~n)mQtcachcluynha'tDi~unay dATIMi co duynha't Vn E Hn thoa (8). GQi ~/(p),1 :$j :$n la cacnghi~mcua(9),Ta co, theo congthdc Cramer, .to 19 Chl/ang2 : Bdi TarinCauchyChaPhztcmgTr'tnhLaplace ~j «p) =!J.j(<p}/!J. ,1~j~n vcri .1j«p)'=det[ail] I$k,l$n va ail «p)dinh biJi a~ «p)= { (gk,gl) khi <Pk khi 1* j 1=j Ta tha'yngay .1/<p)la mQt da thu-ctheo <Pj, , <Pn nen lien t1;1ctheo <PI,...,<pn. V~y ~j«p)tity thuQClien t1;1cvao <PI,...,<Pnva do do vao <p.Tit day ta congayvnL<P)tuy thuQclien t1;1cvao <p. Nhu v~ythu duqcke'tqua (xP1J1[8]) Vai inJi n EN (8)conghi~mduynhdttuythuQcvaorptheonghia£2. Nhu dii noi tren, (5) ne'uco nghi~mthi co vo 86 nghi~m.Trong t~p nghi~m nay co duy' nha't mQt nghi~m,gQi la Vo thoa tinh cha't dV6(1+~2)d~)1/2co tri ghi ct!cti~u.Bau day chungtOi xa'pxl nghi~mnay J bilngcacnghi~mcuabai toanmomenthO\th~n. Trucrche't,chungtOitrt!cchudnhoahQcacham{giheN.GQi{edieNla h~ trt!cchudnduqcxaydt!ngtit cachamgI, g2,..., taco gi el =IIglli n gn - :L(gn,ej)ej j=i en = n gn - :L(gn,eJej j=i ... 20 Chuang2 : Bdi Toan Cauch~Ch~PhuangTrinh Laplace Nhuvl%y (eiLeN IamQtC<1sd trt!cchmfilcuakhong gia~coil dongH cuaL2(J) sinh bdi {giheN H =(gt>g2"") Dodococach~ngsf)Cij , Mij , i, j E N sao'cho Cij, Mij =0 vai i <j va n e. =~ Coo g .1 L.J IJ J j=i n gi =LMijej' ,Vi E N" j=i Mijduqc tfnh theo Cij bdi n "0< Mij =LCjk(gi,gk) k=I ( .> .)I_J D~t Cn =[Cijt~i.j~n Mn =[Mij]l~i.j~n Ta co M =C-In n (nEN) (10) Vai mM day sf)th1!Cq>={q>ilieN ta d~t i Ai(q» =LCijq>j , i EN j=l n vn(q»=LAi(q»ei i=l (11) Thi vn(q»Ia nghi~mcua(6)chobdike'tquacuado~ntren. Thl%tvl%y,vai vn(q»chobdi (11),ta covn(q»E Hn . H<1nuavai Is ks n, n (vn(q»,gk) = LAi(q»(ei,gk) i=I n i = LLCij(ei,gk)q>j i=l j=l n n = LLCijMkiq>j i=l j=l .Ai 21 Chuanggn:Bai TorinCauchyChoPhuangTrinh Laplace n = L8jh:<pj j=1 = <Ph: ~ Tli tlnh duynha'tnghi~mcua(8),ta co (11). Tie'ptheo,g<;1iH* la khong gian trt!Cgiaovffi Hn, Vola mQtnghi~mcua (5) ,thi ta co (PHilVa) E Hn va (PHilva - Va) E H* vdi PHil la phep chie'u th:1nggoclen khong gian Hn, tac la (PHilva,gd =(va,gJ , V'i E {1,...,n} NhtfngVola ;;ghi~mcua(5) : (vo,gi) = <p(Xi)= <Pi(1 s i s n) V~yPHilva thoa (8),nghla la vn«P)= PH van (nEN) (12) OC! DovaEHva {Hn}neNla daytangcackhonggianconcuaH va URn tru n=l mat trong H, Den ta co tlnh cha'tPH va ~ va theo chmin L2 khi n ~ 00 hay, n vdi !>>0 rho trudc, ta c6 no E N saorho II PHIl v0 - val/<~ V'n :2:n a, (13) N6i cachkhac, day cac ham (vn«P)neNhQi tl;]v~Va.Ham vn«P)la xa'pxl thti n cuaVo. , Bay gia chungWi xet truang h<Jpve'phai cua (5) c6 chtiasai s6 (do do d~c,tlnh tOaD,...) va nhu the'bai tOaDc6 thg khong nghi~m.Chung Wi se udc lu<Jngsai s6 giua nghi~m'xa'pxl Vn va nghi~mchfnh xac Vodudi anh huangcuanhi~ua ve'phai. Jo ')') Chuang],-=-l1g,i_ToanC uchychophlfangtrtnhLaplace chudnEuclideva chudnsuptrenRn.Vie't VcJix = (Xl,..., Xn) ERn II X 112 =(xi+...+x~)1/2 II X t =sup{1Xl I,..., I ~nI) " Ta ky hi~uII Cn II rho chm1ncua Cnxemnhli anh X~tuye'ntinh ta (Rn,II.1I2) vao (Rn,II.ILJ' ChQnfla mQthamtangnghiemeachta [ 1 ,00) len [II glll-1 ,00) saDrho , f(n) ~ II Cn II , 'v'n~1 Ta coth~chQnehiingh~n ' f(n) =n-1 +max{II Ck 1111~ k ~n } (15) vcJionE N va f la affine trong tU'ngdo~n[n, n+1]. ;:-Khi d6 f-lla tangnghiemeachta [II gl r1,00) leD [ 1,00) . VcJi 8 E [0,11gl r2) Ta eo 8-112~ IIgill-l nenta coth~d~t n(8)=[f-1(0-1/2) ] (16) GQi Vo la nghi~m eua (5) trong L2 (J) ffilg vcJi ve' phai eua (5) <p= (<P(Xi)}ieN.Thay <Pbai day eacs6 tht;tc11=(111,112'."')thoamandi~uki~n . 111l-<p11~8'" 00 Theo (11) n II vn(11)- vn«p)II =II ~.)~i(~L)- Ai «p)]ei i=l ~llcn 11.1I1l-<pt Jo 23 Chuang2 :Bdi roan,Cf!Y2!JychophuangtrtnhLaplace ~f(n):o V6iu(0) chobd'i(16),ta con(o)~fl (0-112),ughla la, nn(o» ~0-1/2.V~y II vn(o)(J.l)- vn(o)«/»II ~ 01/2 (17) M~tkhacv6inEN ta co II vn(J.l)- V0 " ~ II vn(J.l)- Vn«/»" +"PHnv0 - v0 " L8) Vi fl tang ughiem cach tit (II g1 r\oo) leu [1,00)ta co "" lim f-1(0-1/2)=00 o~O nghiala n(o)~ 00 khi 0 ~ o. Do do co 01> 0 saDcho u(o)~none'u0 < 0 ~ 01.Ch<;m0 < 0~mill {OI,G2/4} ta cougay tit (17). II vn(o)(J.l)- vn(o)«/»II ~ ~ (19) Tit (13), v6i G> 0 chqn nhu d'(19) !!PHnVO-VO" < ~ (20) Tit (19),(20)va (18)ta thu duqc II vn(o)(J.l)-vo II <G Nhuv~ychungt6i thuduqcke'tqua(r8 I) . GQiVala nghi~mcua (5) trongL2(J) zingvai da ki~nchink xacave' phdi rp={qixJh~IN.Vai mQi Ii> 0 cho tntac, luon luon co 0 < 8 < IIg1112 de' chokhi thayv€ phdi cua(5)baidaycacso'thlfCP =(PI, P2,...) (pla da kifn a v€ phdi cochziasaisa)thdadi~uki~n. Ji 24 Chuang2 : Bdi TorinCauchychophuangtrlnhLaplace II ~- cp1100 ~ () thi co II vn(o)(~l)-vo II <E Trang do n(8)dztqccho biJi (16) III. Bai toaD Cauchy cho phudng trinh Laplace tren nna khong gjan. -... 111.1Bai toaD. B:H tmin dt1<;1Cxet la : Tim mQtham u(x,y,z)di~uboa tren mJakhong gian tren Q D ={ (x, y, z)l- 00 a}, lien t1;1ctren D = {(x, y, z)l- 00 <x,y <00 ,Z ~o} , thoa cac di~u ki~n u(x,y,O) = uo(x, y) va au = uz (x,y,O) = Ut (x,y) vai X2 + y2 < 1. Noi cach khac, f)z xet"baitoaDbien: V2u(x,y,z)=0 u(x,y,O)=uo(x,y) uix,y,O)=Ut(x,y) trongdo v2 la toaDtil Laplace.Day la bai toaDCauchyrho pht1<1ngtrlnh Laplacetrennilakhonggiantren. ..~' .io 25 ,(x,y,z) ED, (20) X2+ y2 < 1 (21), , ,X2+ y2 < 1, (22) Chucmg2 : Bai ToanCauchychophuongtrlnhLaplace 111.2.Thie't l~pphtidl1gtrinh itch phan Ky hi~u: K::: {(x,y,0)IX2+ y2 < 1} Q =R2\ K Bai tmin(20).(21),(22)la bai tminCauchytren mlakh6nggiantrenD. Lc1ytri ghibienNeumannv(x,y)=uz(x,y,O), (x,y)E Q la :1nham.Khi d6ne'u Urn dttqcv(x,y~,(x,y) E Q thi se xac dinh dttqcu(x,y,z), (x,y,z)E D. Nhttv~y. . chungta c6bai;,tminCauchyngltqc.Bai tminnay la rnarQngcuabai toaD (1),(2),(3)tl1hai chi~usangbachi~u.Chungtoi thie'tl~pphtt<1ngtrinh tich phan tinh v D~t nx,y,z;~""C;;) = (47t)-1[(x-~)2+(y_,,)2+(z-C;;)2r1l2 N(x,y,z;~""C;;)= nx,y,z;~",C;;)+nx,y,z;~",-C;;) trong d6 r la nghi~mC<1ban cua phttcmgtrinh Laplace va N la ham Neumannchophtt<1ngtrinh Laplaceungvai di~uki~nNeumanntren bien z =O. . Ap d1;1ngdtingth((cGreenII chocachamu va N tren mi~nDE, I':>0, vai DE=D\BE va BEla quac~ud6ngtrongD c6bankinh 1':,tamt~i(x,y,z) r6i choI':~ 0, chungta dttqc,saurnQts6s:ipxe'p. 1 J J W(~,")d~d,, u(x, y, z) =- 27t. [(x - ~)2+(y - ,,)2 +z2J1I2-00 -00 (23) trong d6 Jio 26 Chuang2 : Bdi TolinCauchy~ho_p}y!!mg_t"L1JhLaplace .~ { Ul(1;.ll), (1;.ll)EK , W(~,11)= v(1;,11) , (I;,11)E Q Cho z ~O+, chungta dtiqc 1 J J W(I;, 11)d~d11 u(x,y,O)=- 21t [(x- ~)2+(y,11)2]112-co -co V6i (x,y) E K, chungta dtiqc ~If v(1;,11)d~d11 - u (x' ) - ~If U1 (I;,11)d~d11 21t ~~x- ~)2+(y -11]1/2- 0 ,y 21t [(x - ~)2+(y -11)2]112Q K hay If v(I;,11)d~d11 =(p(x,y)1/2 [(x - ~)2+(y -11)]Q (24) trongdo cp(x,y) =21tuo(x,y) - II U1(~,11)d~d11 K [(x - ~)2+(y -11)]1/2 PhtiC1ngtrinh (24) la phtiC1ngtrinh tich phan tinh v. Day la phtiC1ng trinh tich phan tuye'ntinh lo~i1.PhtiC1ngtrinh nay khong chinh theo nghIa ho~ckhongconghi~m,ho~cne'uconghi~mthi covo s6 nghi~m.Chungtoi sedungnghi~mcuacaebai toan momenthil'uh~nd~xa'pxi nghi~mcuabai tminnay. 111.3.Chinh boa. Cho {(Xi,Yi)JieNla mQtday tru m~tcaedi~mtrong K. T~i caedi~m {(Xi,Yi)J,tit phtiC1ngtrinh (24),chungta dtiqc. .Ai 27 Chuang2 : Bdi Tt)(inCauchychophuangtrinhLaplace ff v(~ 11)d~d11 Q [(Xi-~)2(Yi-11)2]1/2 =<P(xi,Yi), i E N (25) PhuC1ngtrlnh (25) duc;rcgQi la "BM tmin moment" theo Stieltjes. Ham v thuQcvao H la khong gian con cuaL2(Q)sinh bcri{gi}ieINdQcl<%ptuye'ntlnh. 1 2] 1/2 gi(~11) =[(Xi _~)1 +(Yi -11) i EN, (~11)E Q H = Ky hi~u( , ) vllll.11Idn luqtla tlchvohuangvachm1:ntrongL2(Q),nghlala rJ(Q) ~ {2d~d'1<oo} «p,\fI)=ff<p,vd~d11. Q II~II~W ) 1/2 <p2d~d11 ' BM tminmoment(25)duqcvie'tl~ila (v, gi)=<Pi, i E N vai <Pi= <p(xi, Y i) , i E N GQiHn la khong gian sinh bcri{gi},1::;; i::;; n HI) =, nEN va cho<P= {<piJ ieINla mQtday s6 th\!c.Vai m5i n E N, chungta xet bai tmin Urn ..A'o 28 Chll<m/I2 :Bdi ToanCauchychoph,lJdngtrinhLaplace vn=! i=1 ~igi eHn (~l""'~n E R) saDcho (vn,gi)=<Pi , 1 ~ i ~n (26) Phuila bai tmin momenthfiu h~n.Bai tOaDnay co nghi~mduy nh:1tVn tuy thuQclien t1;}Cvao <Ptheonghla L2. (xem II.3 chu<1ngII). ... Nhu da:noi it do~ntren, (24)ne'uco nghi~mthi co vo s6 s6 nghi~m. Trong t~pnghi~mnay co duynh:1tmQtnghi~m,ge;>ila Yo,thoatlnh ch:1t ( ) 1/2 l~v@d1;d'l co tri gia eve W!u. San day chUng tOi xa"pxl nghi~m nay b1lng cacnghi~mcuabai tminmomenthfiuh~n. Ge;>i{ei)neNla he;>tr1!cchmin xay dtJng duqc tit he;>{gi)neNbftngphu<1ng phap Gram - Schmidt,ta covai DEN g1 e1=~ [ ' gn - I(gn ,ej)ej ]J=1 gn = n gn - L(gn,ej)ej j=1 va n e. = LCijgj1 . 1J= n gi= .LMijej' ,Vi e N J=1 vai Cij, Mij la cac hftng s6 saDcho Cij = Mij = 0 khi i <j Jio 29 ChztcJnfl2: Bdi.Toan flauchy~",~phU:(}ngtrln"'--Laplace Vai day 86 tht!e q>={(pd,i e N d~t i Ai(q» =LCijq>j , iEN j=l thi n vn(q»=LAi(q»ei i=l (27) la nghi~m(duynha't)eua(26) -.. GQiH* la khonggiantrt!egiaovai Hn,VoIa mQtnghi~meua(24)thi vai q>=(q>(Xi,Yi)} , 1 ~ i ~ n vn (q»=PHnVo neN (28) trongdo PH)a phepehie'utrt!egiaolen khonggian Hn va khi n ~ 00 thi PHn v0 ~ v0 theoehminL2,nghlala vai mQiE > 0 ehotruae,ta co noe N saDeho II PHn Yo-yo II <; '\in ~no (29) Noi eachkhac,daycaeham {Vn«P)}neN hQit1;1v~ Yo.Ham Vn(D)la Xa'pXl thU'n eua Yo. Xet trucrnghqp da ki~n q>0've'phai eua(24) co ehU'a8ai 86 do do d~e, tfnh toan, .., va nhu the'bai toan co th~khong co nghi~m.Chung Wi uae ht<;1ngsai s6 giuanghi~mxa'pxl VIIva nghi~mchlnhxacVodudicluhhuang euanhi~u0've'phai.Ke'tquad~tduqela. GQiVoto,nghi~mcua(24)trongL2(Q)tl1angangvaida ki~nchinkxaca vi phdi rp= {q{x;,y;)J, i EIN. Vai mQi 8 > 0 cho trl1ac,[w5n[non co .Ai ~O Chuang2 : Bai Toan CauchychoPh,ltf:1ngtrjrthLJ1-place 0 < 8 < IIgll12de?cho khi thay vt phdi cua (24)bai day cac 86 thl/Cp ={pJ, i EN (p chink la da ki~ncochita8ai86)thoadi~uki~n II Jl - (P1100~8 thi co II vn(o)Jl- Vo II trongdo n(&>dzt(lccho bai (16) ".~ IV. Bai toaD Cauchycho phtidng trinh Laplace trong mQt dai khong d~u. IV.I. Bai toaD.Baitmindtiqcxetla : Tim mQthamu(x,y)di~uboatrenmi~nph:1ngD dtiqcd~nhnghlabcri D ={(x,y)1-00 <x <00 ,0<y <«<x)} va lien t1;1ctren D ={(x,y)l- 00 <n <00,0~y ~«<x)}thoa cae di~uki~n (x),au/Ox=Ux(x,(x)= fix), au/OJ= Uy(x,(x)= g(x), u(x, (x»= Ul(X)vai -00 <x <00. N6i eachkhae,xetbai toanbienUrnUl thoa. V2u(x,y) =0 Ux(x,«<x»=f(x) Uy(x,ct>(x»=g(x) u(x,«<x»=Ul(x) , (x, y) E D XER XER XER (30) vdi ,f, g, ulla caehamchotrude,\72la tmintiJ'Laplace.Dayla bai toaDCauchychophti<1ngtrlnh Laplacetren mQtdiU khongd~unltmtren m~tph:1ngtren y >0, cacbien la caedtiClllg=(x)va y =0 (xemhinh 2) Joi n.. Chztang_~-=_BalToa~auchy chophztangtrlnhLaplace IV.2. Phtidng trinh tfch philo va chinh boa. Di;}t f(x, y;~,11)=- 2~In((x - ~)2+(Y - 11)2)-1/2 G(x, y; I;,11)= f(x, y; I;,11)- f(x, y;1;,-11) (31) trongd6la r nghi~mcC1bancuaphtfC1i1gtrlnh Laplaceva G la hamGreen cuatminta Laplacerlngvcridi~uki~nbienDirichlett:;libieny =o. Chi c:1nxac dinh u(x,O)= v(x) thi khi d6 u(x,y)hoan toan dtfClcxac dinh.D€ rut ra phtfC1ngtrlnh tichphantinh v, tich phandiingthrlcGreenII tren mi~nDE ,E>0, vcriDE=D\B'E ,vaB'Ela quac:1ud6ngtrong D ban kinh E tam t:;li(x,y)rdi cho E ~ 0 (xemhinh 2) y =«!>(x) Y 1 0 B'I u: E I I I I I , X Hinh 2 Chungtoi nh~ndtfClCphtfC1ngtrlnh foo yv(~) foo-; 2 2 d~=-u(x, y) - G(x,y;I;,~(~»f1(~)d~+ ~(~-e +y ~ +f00 G1(x, y; I;,~(~»u1(~)d~ -00 ( ~ 00 <x <00 , 0 <y <~(x),) vcri £1(~)= g(~) - ff~)'(~)va (32) .. 1>. ~') Chuang2 :Bai TorinCauchYnchop/1,l{qngt,.tnhLaplace G1 (x, y; ~cjJ(~»=GTl(x, y;~ cjJ(~»- G~(x, y; ~,cjJ(~»cjJ'(~) 1 Y - cjJ(~)- (x - ~)cjJ'(~) = 21t (x - ~)2+"(y- cjJ(~»2 1 y +cjJ(~)- (x - ~)cjJ'(~)+- 21t(x - ~)2+(y +cjJ(~»2 (33) Trang cac tinh taan tren, chung tOi dii sti d\1ngrnQt so'gia thuye't v~ dii' ki~n cha va nghi~rn rnu5n Hrn nhu Bau : Gia thie't (i) <D'(x)= a vai Ix Ikhci16n. ..... (ii) f{x),g(x)va u1(X)tie'nv~a khci nhanh (chiing h~U1nhanh nhti la I;I khi Ix I-+ 00) (iii) (1 +x2)1/2v(x)thuQcvaa L2(R). Cha y -+ <D(x)trang (3), chung ta duqc (xern [Po LattJ trang 144 - 145) 1f00 cjJ(x)v(~) 3 f00-; 2 2 d~=-"2 ul (x) - G(x, cjJ(x);~,cjJ(~»f1(~)d~+ -00 (x -~) +~(x) -00 +f00 G1(x,cjJ(x);1;,cjJ(~»ul(~)d~ -00 (34) dayla phuC1ngtrinh tfchphantfnhV.Chungt6i Bebie'nd5i (34)thanhrnQt phuC1ngtrinh tfchch~p. Chuy dingham f00 yv(~) d!:1 2 ~ H(x, y) = 1t --<XJ(x - ~)2+y di~uboatren ntiarn~tphiingtreny >O.Tri giciH(x,<D(x»chfnhla ve'trai cua(5).Dg tfnh : (x,cjJ(x». chungta dungke'tquatrang [5J : 1 foo (y - C«~»-+ - ') U 1 ~)u., 271:-00 (x - ~)2+(y - c«~»2 271: -00 (x - ~)2+«IX'x)- c«~»~ khi (x,y)-+(x,<D(x». .it 33 Chuang2 : Bai ToanCauchychophuangtr'inh~aplace Bay giCJ: (x,Ij>(x»chfnhla gi6ih~ntitphfadu6id~ohamtheophtfang cuav~phai cua (3)khi (x,y) -4- (x,(x»,Ii la vectadcrnvi phaptuy~nhuang nQicuaduCJngcongy =(x)(Hinh 3) y=$(x) Ii =(Ij>'~~), ~) 1 a =(1+'(~) 2)2n Hinh 3 GQi A(X) =H(x,Ij>(x»,~(x)=: (x, Ij>(x». thi H(x,y)co th€ bi€u di~nnhu mQtth~v6i caem~tde?A, ~ tren mi~n y >(x). Bay giCJchung ta vi~tbi€u thac tuCJngminh cho A(x)va ~(x).Chung ta co A(X)=-! Ul (x) - fXJ G(x, Ij>(x);~,1j>(~»f1(~)d~+-00 +f00 G1 (x, Ij>(x);~,Ij>(~»ul(~)d~-00 ~(x)=tf1(x) -'a(x)-l f00 G2(x, Ij>(x);~,1j>(~»f1(~)d~+ . -00 1 foo (Ij>(x)+1j>(~)W(x)+x - ~ , ()d + 21ta(x)-00 (x - ~)2+(Ij>(x)+1j>(~»2u 1 ~ ~ 1 J oo (Ij>(x)_1j>(~»Ij>'(x) +x - ~ , ()d +2mx(x)-00(x - ~)2+(Ij>(x)_1j>(~»2u 1 ~ ~ (35) ..a:. QA Chuang2 :_BairQrj,nflauchycho_phuan~tdnhLaplace trong do G2(x,cp(x);~(H~»=Gx (x, cp(x);~,cp(~».cp'(x) - Gy (x, cp(x);~cp(~» a(x) = (1 +cp'(x)2 )112 Bd'i(35),cacham m~tdQA(X)va f.l(x)dttgcdinh nghlatren R va tuy thuQcmQtcach lien tvc vao (x),'(x),ul(x), u'l(x), f(x) va g(x) theo nghla L2(R). Bay gia, tlch phan d:li1gthac Green trong mi~n 0'.- DR ={ (x,y) Ilxl<R, cp(x)<y <R } vachoR ~ 00 , chungta dttgc H(x,y) =Jex)nx,y;~,CP(~»f.l(~)d~--00 - J ex) r1(x,y;~,cp(~»A(~)d~ -00 (36) vai - 00 (x)va r1(x,y;~cp(~»=r~(x,y;!;;cp(~»cp'(~) - rf1(x, y; ~cp(~» 1 (x - ~)cp'(~)- (y - cp(~» =2x (x - ~)2+(y - cp(~»2 chuy r~ngkhi R ~ 00 , tlch phan tren CR ={ (R, y) I ~(R)<y <R } u { (-R, y) I ~(-R)<y <R }U {(x, R) 1-R <x <R } tie'ntai 0 doht:quacuagia thie'ttrenv ( nghlala, (1+x2)1/2v(x)thuQcvao L2(R) ) Tinh tri gia H(x,y) t~i (x,k) vai k la mQt s(f c(f dinh Ian han (x)vai ffiQix trong R, chungta cobd'i(36) .1fex) kv(~) f ex) x -ex) (x - ~)2+k2 d~= -ex)nx, y;~cp(~))f.l(~)d~- - fex) r1(x, y; ~cp(~»A(~)d~ .-ex) ..foi 35 Chu:ang-~~l1r!L'[o2-nCauchychophu:angtrinhLaplace GQi F(x) =1tJ 00 r(x, y;l;,4>(~»Jl(~)d~--00 - 1tJ 00 r1(x, y; l;,4>(~»M~)d~ -00 thi chungt6i nh1%ndttc;rcmQtphtt<1ngtrinh Hch phan d~ngtich ch1%pHnh v nhtf sau J oo kv(~) d~=F(x) -00 (x - ~)2+k2 , \/x E R (37) Dayla PDtf<1ngtrinh Hch phan lo~iI va chungta bie'tding bai tmln nay la kh6ngchinh.Chungt6i sexay dt!ngmQthQ(vr3),r3> 0 cacnghi~m chinhhoa(xem[35]),va chungt6i chQnra mQtnghi~mchinhhoa"g~n"vai llghi~mchinhxac.Nghi~mchinhhoa la mQtham6n dinh d6i vai st!thay d6ia ve'phaicuaphtf<1ngtrinh (37). IV. 3. Chinh boa. D~t. k 5- 2a(x)=v-;x2+k taco a *v(x)=k f00H ~ 2 v(~)d~=F(x) 7[ -00 (x- ~) +k hay. a *v =F (38) ta tinh dttc;rc a(t)=e-kl t I , (t E R) trongdo a(t) la bie'nd6iFouriercuaa(x) J;. on Chuang2 : fjai 'J'Qa1J~(J,lf,chychophztangtrlnh LaI!lace &(t) =k 1:a(x)e -ixtdx .2 1,1 =- . Vai v trongL2(R),chungta cotit (38) &(t).v(t) =F(t) (39) (39)khong chinh vi v(t) =f<t)I'&(t) khong 6n dinh. Chung toi dung phu'dng phapTikhonovd~chinhboa. Vai mip>0,ham .... 'P(t) =&(t)(p +&2(t»)-1F(t) (40) thuQcVaGL2(R). D~t vJ3(t)=k 1: 'P(t)eixtdt (41) thi Vp E L2(R) va, bai (6),Vp thoa phu'dngtrinh pv J3(t) +&2(t)v J3(t) =&(t)F(t) \ft E R (42) va tuy thuQcmQtcach lien tt,1CVaGF(t). Khi f3--+ 0, fa dll(/Cupcho biJi (41) la nghi~mchink h6a clla (37). Bay giCr,xet tru'Crnghqp v~ phai cua (37) chu-a8ai 86 do do d~cho~c t.illh tmin.Vai nhi~uav~phai,(37)coth~khongconghi~m. Sandaychungtoi u'aclu'qng8ai86giuanghi~mchiilh hoav"va nghi~m chfnhxacVodu'6ianhhu'angcuanhi~ua v~phaicua(37) Bay giCrgiVoEH1(R)-langhi~mchfnh xac cuaphu'dngtrinh .. OM Chu:cmg2 : Bdi Toan Cauchychophu:cmgt"Lnh~_aplace a(t):vo(t) = A F0(t) 'itER (43) v6iF va FatrongL2(R)saorho IIF-FolI<E (44) H1(R)la khonggianhamchli'acaehamf co d~ohamf saochof va f binh phll<1ngkh:i tich. Tit (42)va (43)suy ra J3(Vp(t)- vo(t» +a2(t)(vp(t)- vo(t) =-J3vo(t)+a(t)(F(t) - Fo(t», 'it E R (45) -... Nhanhai v~cua(45)v6i (vp(t)- vo(t») r6i tichphantrenR, r6i rho J3=E,suy ra (xem[5]) lYE -Yo 12~Ivo 12+(~)1I2 (46) Tll<1ngtt!nhanhai v~cua(45)rho t2(vp(t,)-vo(t,»va tich phantren R, r6i rho J3=E.Soy ra ([xem[5]) IIt,(ve-vo)II~IIvo II+e-l(~)1/2 (47) VI IIve-voll=IIve-voil va IIv~- Vo II=II t(,ve - V0) II Hi (46)va (47)chungta dllqctheo dinh nghla cuachua:nHI, ky hi~uII. "Ht II vE - Vo IIHI"=II vE - Vo 11+11v~ - Vo II~ Kl (48) trong do KI =max (!!voII+(i)1I2,IIv'o II+ e-l(i)1I2) Chung ta co,v6i t" >0 ba'tky, -'" 38 Chuang2-: Bai ToanCauchyChoPhuang_'!,,:ll1h,Laplace J I ve(t) - vo(t) 12dt ~ It I:te J e-2kltle2ktelve(t)-VO(t) 12dt. It I:te foo 2~ 2n-1e2kte &2(t)1 Ve(t) - vo(t) I dt -00 2 -1 2kt 1 h ( h h ) 1 2= n e e exv - Voe 2 ~ 2n-1e2ktcK1 E(I V 0 12+ (~)1I2) K 2kte= 2Ee (49) trongdo K2 =2n-I(1 Vo 12+(~)l!2) VaiE < 1, chungta co theo[5] t;l ~2(1+k)(ln(~»-l (5: B<1i(49),(50)va (51),chungta co II vI':-VO112~2K2t;2 ~K(ln(f»1/2 trongdoK =8 (1 + k)2 K2 Chung tai du'<;1cke'tqua(da rang b6 <1([5]) Gia sitnghi~mchinhxacVocua(37)tu'crngtingvai Fobellve'phaithuQc< vaoHI(R) va rho IIF-Foll <E 11.11=chminL2(R) Thi comQtnghi~mchinh hoa Vccua(37)saDrho II Ve - Vo II ~K(ln(~»-lkhi E ~ 0 Trong do KIa mQthAngs6 chi pht;thuQcvao chminHI cuaVo ..to ~9 J, ve(t)-VO(t) 12dt Jlt12t;21 ve(t)-VO(t) 12dt It I>tc It I>te (50) t;2foo I t(ve(t)-VO(t»12dt=. -00 K t-2 K t-21 e 2 e