Luận văn Một số tính chất của hàm điều hòa

£' '/' '0 ,~, ~l.({(ht Yrl;/& (tJao Cf9c .200/ 36 ,//' '" (i7;' CY/':;c/~{t;JIc/& ,,//ItNt;f Ft? Chu'dng 4 ., , GIAI PHUdNG TRINH LAPLACE ~, ,., ., ~ TRONG MIEN NGOAI CUA QUA CAU ., ' I. Ma DAlJ: Xel b,li l()(1ngialri bienDirichlelcuaphlWngldnhLaplaced6ivdi mot mi6nD (lIdl1lienvabi ch~n): LlU(X)=0, \iXEW\i5 HeX)=rex), LlXEaD tronglit')r 1:1hams6lien t~lctrenaD. -DE chC(ngminh gl)n ma khong mat llnh l6ng quc1t,trongChl(elngnay chungloj xel mi6nD la quadiu deJnvi B={XER": I x I < I} vdi bienla a13={ x E R": I x I < I} trongkhonggianRII vdi n~2. -Bai loan gia lr! bien Dirichlet mi6n ngoai se c6 nhi€u nghi~m.Trong cac Sc1chgiao khoa,ngl(oita thl(OngthemtlnhchatCI}th~cua u 0 vungvo q(c lhl mdi xac lIinh dl(l}Cbi~uthac cua u.Trong chuangnay ,chungtoi l1l(afa ml)tdangnghiQI11u1ngqm1tdll cIll(a1'0tinhch~flcua 1I0 vo c~(c. -N(ji dungcua dll(ling nay dlCl;5Ctdnhbay d~t'atheobai ban IS]. II. lYIQTs6 DJNH NGHIA vA DJNH LY . 1.Dilll, if' : Xel mi2n0 =13(0,1') \ {O}={x E nn:0 D. Khi lIl) lc1nlc.lih~ngso c c6 caetlnhchatsall : . C E (0, I) . Ml)i h2\111c i6uhoadlCOngu xac l1inhtrenQ d€u th6a \ix, Y E 13(o,~), Ixi=Iy\=>clI(y) < u (x) Clul thich: Hang sf)c khongph\!thuochamdi6u boa u. Clllfllg mill" - Gl)i K IA m~tceiubanldnh ~ (tac K =~S) lhl K la t~pcol11pacttrong O. Theo l1inh1)/Harnack thl tl1nt~\ih~ngs6c E (0, 1)co tinhchat J;L(1It 'jrt~t-(&to dt;c 200/ 37 Jfiu;}IbJt- ,9'h{Wth-% C U(Y) <U(X) VtJi nH}ih~lIndi~llhoadl((jngII xacdjnhtfenQ vav6ix, Y E K. - Cui t lllY Y thuoc(0, 1)vau 13hamdi~uhoadlf0ngtfenn. H~l1ns(1 x -1- u (tx) cung 13ham di~uhoa duong tren n, do d6 eung thoa batdanglh(ic0 tren,t(icla CU(ly)<u(tx) , r '\Ix, y E - S 2 M6i x E i S setl(ong(ingduynhatmotph~nto'tx E ~.S, do d6 tr eu (y) <u (x) , '\Ix,Y E - S 2 - Ta thay ~ c6 lh~c6 gia lrj bat k5'trongkhoang(o,~) ne'ut dtiQC chqnthichht.iptrong(0, I). Do d6 '\Ix,Y E B (o,~), I x I =I y I=>eu(y)<u(x). 0 2) Villh If': .Gi~isli II : Rli -1-Ria hamdi~uhoatrenB\{O}={XERII: 0< I x I <I}. . Ham sf)A[ullren B\{O}ducjcd1nhnghlanhl(sall: Vdi x tuyY thuQcB\{O},A[uJ(x) la giatr1trungblnheuahamu tren . m~tdiu b,1nIdnh lxi,tuc la Alul(x):= 111-l fu(y)dS(y) mix! lyl=lxl twile AluJ(x):=~ fu~xl~}lS(~) 1~I=l trunglit)w la di~nrichcuam~tdu donvi. . Khi l16AluJ cobi~llthuenhusau: --. Tncc'lnghejpn =2: .Yiui /1 'Prl~t'~o ~~c 200/ 38 ./1j;?J/Zn ,9'~tV/t~'r:i Alu!(x)=b(-Inlxl)+c, VXE B\ {O} trungd6 b, c IA h~ngso. - TnCongh(Jpn>2: Alul(x)=b IxI2-11+C,VXEB\{O} trongdo11,c IAhhngso. CIll£llg lIlillh Xet hamsof : (0,1)~ R dlC<JCxacd!nhbdi Vr E (0, I), ref)=fll(r~)oS(~) (1) s Saudaytasettmbi~uth((ccllaf( I x I), r6i suy ra bi~uth((cCI1aA[u](x). - Ham s<1-:,~u(r~) lien t~ICd~ukhi r~thuQct?P compacttrangB\{O},ur 00 d6 d~1Ohamcua f IA 1"(r)=f~u(r~)oS(Oor s = f~. ('Vu)(r~)oS(~) S D~lbienmoi~=r~thl dl(\jC 1"'(1')=rl-l1. f;.('VlI)(~)dS(~) rS - Cui rl), 1'1tily Y thuQc (0, 1) l (/z .~o L/;/, J(t(t/{ >(I/It-l tJao O~9C 200/ 39 . Jt;~'JI&l/3h(Mt~CY;i Gqin ={x E Rn : ro< I x I <rd Ap dl,lngcongth(I'CGreen, tadt((}c f U.VlI LiS:::: fllll dV 80 0 trongd6 .an =roSu rl S J - ~ Bell ~E foS ro'U= 1 1 neB ~E rlS rl ( U lAphapveclddrinvi hudngngoaicuaan ). ./'>U=0 (dou la hamdi~uboa) Suyra f =-~.(V'lI)(~)dS(~)+f1. (\7u)(~)dS(~)=0~) ~ r()S 1'1S Do d() f I~ .(Vu)(~)dS(~)::::f l~ .(Vu)(~)dS(~) , 0 S I ~,s ~ Tl'(dJng lhereli~ntren,ta suyfa 'Vr E (0, I), f~.(Vu)(~)ds(~) =h~ngsO' (h~ngsO'nay c1u~:!cgQila kl)r rS - 1'1:(bi~u th(fc clh 1"(r) (j tren, ta SHYra fer) =k1 rl-n, 'Vr E (0,1) Do d6 J k[In r +k2 neu n::::2 1'(r) ::: 1kIr2-11+k) neu n> 2 Ma Alulex)= ~f(lxl), SHYra Alul (x)c6bi~uthacnhtI'phatbi~u(rang0) c1jnhly 11.2. (p '/:'(0 c.iJ/' ,J..u(h, ;/(/;/t l-5ao Ol9C 200/ 40 .f ~ - ('::5,- o;/~ './(p';jIMb .!3hCVlth-f/tt 0 3)Djllh Ii : Gic1slthamsoth~(eu cocaetinheha'tsalt: i) Lila hamoi~uboatrenB\{O}={XERI1:0<1x 1<1}. ii) Lila hamlien t~letren:B\{O}={XERI1:o<1xl S';1}. iii) u(x)=O, '\IxE8B={XERI1:lxl=1}. iv) u(x)2:0, '\Ix E B(O,r)={xERI1:0<1x I <r} (vdi r la h~ngsothuQc (0,I ) ). Khi doucobi~uthuenilltsalt -Tn((1ngh<;1pn=2: u(x)=k(-lnI xl), '\Ix E B\ {O} trongdok lah~ngsokh6ngam. -Tnf0nghejpn>2: HeX)=k( 1 X 12-11- ), '\Ix E B\ {O} lrungdok lah~ngsokh6ngam. Chl'tllgmillh G()iAlul(x)lagiatrjtrungbinheuautfenm~tdu bankinh1 xI. Ta seeh(tngminhu=A[u] trenB \ to}. Xel mi~nB(O,f), rheadint ly 11.1(ehudng4) tontc~ih~ngso eE(O,I) saGchovt'iinH,>ihamoi~uboakh6ngamu trenB (0,f) o~uthoa '\Ix,y ~B (o,~) va I x I =I y 1.cu(y) <U(x) D~l W =u- e.Alu] Chltllg ll1inhdW1echiuthanhcaebttdenlll! salt. Ihioc 1:(Chungminhw kh6ngamlfenB \ {O}) . Coi x E B (O,~) '\IyE B (o,~), Iy I = Ix I =>eu(y)<u (x) , I. . (/) ..,y' ,'fWill f/tZl? (.;YfO ekeD 200/ 41 LA~~:yZlt .9'h(l//bti'f/;i SHYI'd cAlul(x):::; u (x) lI(X) - cAlul(x) ~0 w(x)~0 Doell)w khC>ugamlrenB(D,~) .G(.>iL ==inf {w(x): x E B\{O}} l'{)nlai day {ad lwng B \{O}saDcho w (ad -7L khi k -7 00 Day [a"J BallI trungt~pcompactB nen t6n t~1iday con {bd hOi l~1v~b E D Tn('1ugIH.fpbEB\{O}: Ilam di~uboa w d'.llclfc li6u khi x ==b, lheonguyenIy c~tcti6u lhl w la ham hang lren B\{O}, w c6 gia ld khong am trang 13(o,~) nen suyra L ~O. Tn('jngIH.fpbEGB : w(b)=DnenL =D. Tn(Ojugh(}pb==0 :w khC>ngamlrenB(o,f) nenL ~O. Ta 11IC>ud) L ~0, do d6 w khongam lren B\{O}. V~y la Cl) lI(X) - cAlul(x) ~0, '\Ix E B \{O}. Ihide 2: (ChCrngminh HeX)- Alul(x) ~ 0, '\Ix E B \ {O}) (1) - Cluj \ lll: If!clays6 dlfC}Cc1jnhnghlabai lu ==c 1,==c+lu(l-c) . ......................... IIlI ==c + llll._1(I - c) (2) Tl( l" ==c E (0, 1) la SHYHI lJ > 0 va tl < c + 1 (l - c) ==1, tltc t[ E (0, I) Bang ljUY IH,'P,la c6 till E (0, I) vdi ll1l)i Il1 EN. Ta U) tlll - tlll-J ==C (1 - till-I) >0,dod6day{tm}tang,d6ngthaiclaynay hi ch~nbiji I Henhoitl,lv~gidiIH,lnmatagqi la1. Cho 111.--)-00,ll( (5.2) SHYra I ==c + t (l - c) ;; "'/) ',,/',~jtf(/( fUll '(Jew ,)1oc :..!{,i(J/ 42 . /(y; ~yg1t.?-ZUltlt'Yci l =1 v~y tIll ---)0 I khi m ---)00 . (3) - Xel day ham {wm}yoi Will=U -tm A[ul ' Ta lh,l'y Wm- cA\wm] =u-tm Alul-- cA [u -lmAlu]! =II - tmAIuI - cA Ill! +rImAIIII =u-lc+tm(l-c)IAlunl =U-tlllt-1A\Llml =Wm+l n'rcla WII1+1=wn-cAlwnl - Thcl) kel qLl21(I) d bt(dc I thl hamdi~Llhoa Wokhong am tren B\{()}. Ap dl,lI1gkc'tLlU21(I) Jai vdi hamW()thl Ol((,iCtinhchfltkhongam clla hamWI tren13\{O:, Bang Lluy n"Lp,ta suy ra ham Wmkh6ng am tren B\{O} vdi mQi so' nguycndll\!ng111, tLtCla W11l(X)= u(x) - tmAlul(x)~0, \imE N ,\ixE B\{O}. Clio Il) -} 00thl tm---)01 (uo (3)), suy ra u(x)-AllIj(X)~O, \ix E 13\{0} IhiO'c3 : (Chang minhu =AluJ) (4) Giil sLoe It'\nti;liXu E B\ {O}san cho u(xo) >Alu](xo) Do u lien t~IC[(.IiXonen t6nli;limOtIan c~nV clla Xosaocho HeX)>A[u](xo) , \iXEV, M~lkllac,theoketqua(4)d bt(dc2 taco lI(X)~Alul(xo) , \ixEoB(O,lxol) SHYra gia trj trungblnhcua u trenm~tdtu 8B(0,lxol) iOnh(inA[lIl(xo), llrc la AIul(xo)>AI ul(xo) Dietl n~IYvo 19. Do lit) khongc6 Xol1aOthuQcB\{O}maco trnhchfltU(Xo)>Alul(xo), ket h(}pydi tlnhcha't(4) tadt«}c. HeX)=Alul(x), \ix E B\{O} I L' , 1" (// l:;Y~ .Lw( It Fltlto O{tO (7i9c 20{)/ 43 Jf:f~VJlb;toL9'hCl/lth-%; Ihifie 4 : (ke'llu~n) Alul c()bi~ulhlrc nhl(lrongdinh 191I.2(d1lWng4), nen u Lungy~y. - Tn((Jng hl}P Il co:2: Ta Ct)u(x) co: be-In I x I)+ c, \lXE B\{O} Khi Ix I-~ I lhlu (x) -) 0 nen suyra C=o. Khi x E 13(0, r) lhlu (x) ~0, suy fa b ~0 V~yu(x)=b (-In I x I)ydib ~0 - Tnf(ing hl.ipn >2: Tau')u(x)co:blxI2-n+c, \Ix E 8\{0} Khi Ix 1-) I lhl u (x) -) 0 nen SHYfa c co:- b, Llu lk) u(x) =b ( I X 12-11-I) Khi x E 13(0, r) lhlu (x) ~ 0, SHYra b ~O. [J 4/Di"" /y : GiJ sLYlulll I la Llaycaehamdi~uboalfen l~pm6Dc U'\daynayhQi It.!d~uvC:h~lmsoulfenm6il~pB(a,r) c D. Khi lit')u lahamaieuboalrenD. Clll?llg milllt Cui a IllY yIhuQcD T6n 1<.liquacau B(a,f) c Q 'I'as0chl"rngminhu la hamui~uhoalren8(a, f). Do phepIjnhLie'nkh6nglamlbayd6i Hnbdi~uboacuahamsonenta c()Ih~gd sera=0 makh6ngmilltinhl6ngquat. Thc!) c()nglh(rcPoisson(dinh 19II.l-chlWng 1) tac6 Um(x ) co: fH(~,x)ul1l (~)dS(~), 1~I=r '\lXE 13(0,f), \1mE N Ta Ihfiy ,: 'f ' co '-,,/'.Jtt(ilt /tl/t C:7aoClt~)C 200/ 44 Jt;;~vyJ1b3ha~th-CP;i Ulll (x) - fH(~,x)u(~)dS(~)S;; fH(~,X)IUm(~)- u(~)ldS(~) 1~I=r 1~I=r Coi E >() ILlY Y, do UIl1hOit~1c1~uv~Unen 16nt~lis6 nguyenN thoa m>N=> (IUIll(~)-u(~)I<E,\f~EB(O,r)) Luc utI Do c1()LIlli(x) ~ LIlli(x) - fH(~,X)U(~)dS(~) S;;E fH(~,X)dS(~);;:oE 1~I=r 1~I=r fH(~,X)U(~)dS(~),khim~ ct:) 1~I=r fH(~,x)u(~XJS(~), \fx E B (0, r) 1~I=r u(x) ;;:0SUY ra Do lit) Uui~lIboa tren B (0, r). Suy ra u ui~lIboa trenB (a, r). V~y LIdieu boa Iren Q. 0 5/ Hill" ly : Gia Sl(LUmla chu6icaehamdi~uboatrent~pm0Qc R'\ehu6inayhQi" t~ll1~1Iv~hams6u trenm6it~pB(a, r) c Q. Khi de)II 1fthamc1i~uboatrenQ. CIll?llg 11lillh Binh Iy naylah~quaeuadint Iy 11-60tren. , ~' ':" III-PHEP BIEN DOl K. 1)Binh nghia: .Xet l~pEc R'\{O}. V()ix la motphftntii'toyyclla E,phc1ntll x* du'<;jedinhnghlanhl!'sau __'fa'illi f {[~t«(0.0 (;7{~c20()/ 45 t . - cv CY/.'<-/'jtt;:jlbH~h{1dt4 j/ ,t: x*=~ Ixl2 T~pE* lh(.JcoinhnghTalaE*={x*: XEE}. 2)Tfnh chfttciia x*. 1 a) '\;Ix:;eO,Ix *1=N b) '\;Ix:;eO,x**=x c) '\;lEe R"\{O} , E**=E d) NeuxEE*thIx*EE. e) Neu E la t~ph(.jp {XE RII: (k I x I <I} thlE* la t~ph(fP {xERII:\x I>JI. Cilling lIlinh x I 1 H) I' 'I = IH' = N x > b) x**:=(x*)*=x* _Ixr I * 1 2 - T =x x-' Ixl2 c) Suy 1'atu tint ChEllb). d) Cui x toyY thuQcE*.T6nt~iYEE thoax=y*. Suy ra x*=y** ,maLheDtinhchat(b) thl y**=y,nenx*=y. VGy x* LhuQcE. e) Suyfa lu ojnhnghTacliax* vatinhchat(a). 0 3) Hill II IIghia. Ghl Sl(L~pE e R'\{O}vahamsou : E-7 R . Ham s6 KI ul au\icdjnhnghTanhL(salt Kluj:E*-7R \;IxE E*, K[u](x):=Ix12-11u(x*) (j:' '/,"/) ~~/' ,cL,{ui/l ;/(1/I" Ciao O~~c :::!()O/ 46 ,//~ - 6,- CY/,~ JlJtttJlbll- ,!3h{b714f/c'i 4) /Jill" /y. Gi~isu' Ii c IC\{O}va hamso LI:E~ R . Khi lh\biend6iK cuahamsoKluj chinhlau ,tL1'cla KI Kluj]=u. Clll~ngminh Do E**=EnenmienXclcdint cllaKIK[ull ehinhlaE. 'IIx E E, KI KIuII(x):=Ixl2-nKlu](x*) 1 1 2-11 1 "' 1 2-n ( ", ", ) ( )=x X'" ux"""=ux V~yKIKlull=LI. I] 5)/)i"" /y. Gi<iSlt'E c R'\{O}vahamsog:E~R Nc'uw=Klglthlg=Klw] CIll?llg millh Gia si'i'w=KIgJ thl w :E*~R . Khi Lh)Kiwi c6mi~nxacdinhlaE**=E. Vdi x lllY ylhuQcE,la c6 I 1 2-11 ( ) I 1 2-11 I 1 2-11 ( ) ( )Klwllx)= x wx*=x x* gx**=gx V~yKlwl=g. [J 6)ViII" /y : Gi~lSlt'udieuhoa trenquacauoejnV!B c R'\lien t~lCtrenB. Khi (It)KIuI dl(JCxacdinhtrenRU\B b(~ic6ngthue ~ f Ix12_1 KllIl(x)= ~ "' IQ=I!x- ~I"u(~)dS(~) u(x) . neB x ERn \ B neB Ixl =1 trung (M (I) I;) di~n tich clla m~tdiu don vi . ( ,( X' ("L? -..i;/' ...Lad/( /(blt 0uo ilt9c 200/ 47 ,//' - oT cy/.: J':ft':f/&t- J~(V1t;; }//i CJlll'Ilg millh ThL:o cc)ngLht'fcPoisson Lac6 I I 1- x- ~-u(l;:)dS(l;:),\/ x E B. Il;:- xll1 u(x)= ~ J()) I~I=I ma khi I ~I=I Ihl Mi€n xacdinhciiaKIHIla U!'\B (doHnhchc1tIII.2-chl1dng4). -Khilxl=ILhl KluJ(x)=1XI2-IIU(X*) =u(x) -Vdi X IllY Y LhuQcRI1\B , Laco 11 1-11 KllIl(x)= x - u(x*) 1 I * 1 2 1 1-11 - X=- J Ixl- u(l;:)dS(l;:) ()) Il;: - x'If Il;:l =1 1 1--- =~ f IxI 2-11_lxI2 u(~)dS(~)(0 I 111 I~I=I ~- --~- Ixl2 l~e_=1 =~ J Ixl2-n Ixl2 . u(~)dS(l;:)(0 In 11;1=I Ixll"l;\xl-I;1 =~J Ix12-1 u(l;:)dS(~)0) 1 m I~I=I~Ixl-.-~ Ixl x ~Ixl- - =Ix- ~I Ixl 2 X I ? (do 1l;:lxl-N =Ix-l;:I-) nen II?I x --I KllIl(x)= - J u(l;:)dS(l;:) (O1l;:1=IIX-l;:ln 0 ( L' 'I' /? ~);/' .1Wilt j/~ht' bao (j~(JC .2001 48 JJ:f;':fIblt ,-3Y;;a7t~% 7)J)i"h Ii . Gia sl'rt~pm0E c IC\{O}vahamsou:E-t R . Khi Ll6,hamso1Idieuhoa(renE ne'uvachlne'uK[uj dieuhoatrenE* Cillfllg millh -BliO'e1: Giii sl'r1Ilahamsodieuhoatrenn.11.TachCrngmintKlu] la hamso dieuboatrenR'\{O)nhLrsau. Coi x lLIYy thuQCquadu donvi B. Ta cti u(x) ~ ~ J I-lxi' u(i;)dS(~)~ JH(~.x)u(C)dS(~) ffile:l=llx-ct . 1e:1=1 Theo LlinhIy 111.6(chLWng4) ,vdi I xl >1tac6 Klu}(x)= f-H(~,x)f(~)dS(~) 1e:1=1 Suy ra ~x(KllIj(x»= f~x (- B(e:,x))f(OdS(e:) 1e:1=1 =0 (theochdthieh0 Il-2-d1LWngI) Do LIt)Klujla h~lJl1dieu hoa lren n!\B . Theodint nghla,tac6 'IXE R"IB. K[u](x):~Ixl2-n{lx~2J . Suyra Klulla hamgiairichtrenW\{O}(dorichvahlJpcae h~\111giiii richla hamgiairich). Dod6ham ~(Klu})la hamgiili tiehtrenUII\{0},dongthaiham naylrit;ltieulrenR'\B,suy raD.(Klu})lri~tlieu lreneelR'\{O},tue hamKlu}aiel!boalrenR'\{O}. -Ihide 2 : XGtu 1[1hamdieuhoatrenE c W\{O}. Cui b tLIYY thuQcE*. Ta c6 a=b*thuQcE thoa b=a* . TheodjnhIy IIl.6-clutong1,t6nt~\imQtIanc~nVacllaa saDchoue6 thekhailri2nlhanhmQtehu6ihQit~1delltrenvunge~nnay. ,/' . I. '-/;J ~:i;/' dWi/( ';/(;;jt.'c5ao (j~9c :!(/(J/ 49 ' j~;?JfZ'lt-c?hCVlth~i Do CD 00 HeX)=LPm (x- a)=L Um(x) m=O m=O Lrungd6 moi UIIlla da lh(tcdi€u boa . KlulIll(x)= Ix12-1I1I1Il [ ~2 J vachlloidalh(tcI Um Ixl III =0 hOil~ld€u w Lren Va ,Lasuy ra c1L((,icchlloi LKlull1} hOil~ld€u lrungmOlHinc~nclla b, 111=0 khid6 ,~KllIm}~{~Um ]~ Kill] Hams6Klulla tC1ngcuamQLchlloicaehamK[ulll},maCell:ham KllIlIl}dicuboa lrenE*c R'\{O}(kelquabl(OCI), nenlheodinhIy II.6 (chuoung4) Lac6Klu J la h~\Jndi€u boalrenR1\{O}. IhiO'c3 : Gj~1Sll'KllllIA hamdiellhoa,dou=KIKlulJ nenthenketqui1cllabuoc 2 Lac6 1II,}hAm c1i€lI hoa , LJ CJlIi lhich: . MQLpluwngplu-1pkhacd~changminhdinhIy Lrenla dlfaVaGbi~u th('oK[uI(x) ~ [x12-n 1I( [X~2) di!tjnhcael1~nhamdongphilncuaK[u] then cacd~loh~\Jl1riengphgnCllau,saud6 tinh i'1K[u]va changminhbj~uth((c naybhng(), [V-NGHI(~M CUA HAl TOAN DIRICHLET MIEN NGOAI. lIBjllh Ii : Xet bAiLm1nDirichletd6ivoimienngoaicoaqU(1diudonvi B trangRI1 (vdjn>I) i'1u(x)=0, '\Ix E R"\B HeX)=rex),'\IxE oB Lrongl1{)r lient~lCtrenbienoB vanghi~mu E C2(R"\B) n C (R'\B). MOLnghi~md~cbi~Lellabailoannayla (/' ,y' (£' ~iI." j.fUj/{ f/ rl/~ 6ao cree l!OO/ so .jfj.;:y&~5YhMA% j~.flxl2 - IU(X)= 1 w I - I II f(~)dS(~) al3x ~ t"(x) neu xERII\B neu x EoB lrung(16w ladi~nlfchciiam~ldu donvi . C/lllllg l1linh XCl hamv tren B nht( sau ~ f l-1xI2. v(x)= I WaBlx- ~rt(~)dS(~) nell x E B rex) nellx E oB Theoc()l1gthCrcPoissonlhi hamvEC2(B)nC(B) vav dj@lIhoatfenB. Hamv xacdjnhlfen B\lO} thlhamKlvi xacdjnhtrenR'\B. TheoL1jnhIy llI.6-cht(dng4 ,K[v] cobi€u th(rcsall {--HlIllv di@uhoatfent~pmdB\{O}nentheodjnhIy 111.7-chuang4 la c6 Kl vIl~1h~lIndi@uhoa lfen R"\ B . -D€ changminh KI uI lien l~lclfen R"\B, ta chI dn ki€m tinh lien t~IC cib Klvll<.li013. Cui ~lllY 5'lhuQcoB . Neu x -7 ~trong R"\B lhi x':::;-;- --j-~:::; ~ :::;~, hk dov(x*)-7f(~). Ixl- I~I I 11 2-11 ( X J Jim Klv](x)= lim x v ---;; =f(~), x -7 ~ X-7 ~ Ix/- Do LV) v ~yKI vI lien t~ICtrenI~"'\B. Ham sf{Klvl thoacaetlnhcha'tciia b[tiloan nen la mQInghi~meila b[li loan. 0 I f!xl2 - I Klul(x)= 0) Ix-111 f()dS() A'neu x E RI1\ B II=I rex) nell Ixl=1 ,5.£:(2/1CPa~~«(feLOc1l~c !!!{/o/ 51 ,/t:Y~t;!I&~9'h(Mih% 2/ Dillh Ii : Xet bAiloanDirichletdoivdimi~nngoAicllaquadu dejnV!B trongRl1 (vdin>1) L1U(x)=0, \:Ix E RII\ B HeX)::: rex),\:Ix E 3B trongd6r lienl~IClrenbien3B. Khi d6,mqinghi~muEC\R"\B )nC(R'\B) clla bAi loand~uc6 d<.tng lrennll\B nilL(sau [ ] ) 2--11 X 1 x--l u(x)::: I x l w ~ +- III r(~)dS(~) Ixr CD3131x- ~111 , \:IXE n!\B trongde): w Iii l11e)th~1111::;6di~uboa lren B\{O},lien t~ICtren B \{O}va c6 gi<lt1'1 trenbien 813b~ngO. CDI~Idi~n llch clla lI1~lcall (jdn vi. CJul tbieb: .Trong l\"L(Ongh(}pn=2thl Uco lh~Vie'ldl(oi di:.\I1gsan I 1 2" 1'2-1 u(1',H) =w(-,8)+- J ? f(p)d(p l' 2rc 01'- +1- 21'cos(8- (p) ,Vel',8) E (1,CX))x(O,2rc) CluIng millh Ta lh{{yh IA II1Qlnghi~mclla bAi tOi:ln(lheo djnh 19IV.l d lren). Gia Sl(II lA mQtnghi~mba'tky cuabili loan dangxet. f)?l g(x)=u(x)-h(x) Ta Cl)g Iii mOl hamdi~lIboa tren R"\B ,lien 19ct1'enR"\B,c6gia td 0 lren bien aB. J f()dS() '" x E n.11\ BneB -Gqi h(x)= CD1I:::llx - 111 '" Ixl:::1rex) neB (? .'j,' (a c".' .L(trilt F Ii", lDaD?(9C 200/ 52 Jt;~~JJt-3hlVlb~% (:;l)iw=Klgllhl w X,1c((jnhtrcn B. I-Hling di~llboatn~nt~pmo' R"\B nenw la hamdi~llboatfenB (do dinhIy IIl.7-cllltdng4).Hamg b~ng0 tfenaB nenhamWcungb~ng0 tren as. -Coi ~ILlY Y thllC)CaB Nell x -+~tIll x'=-;- --7-; = ~ =~,d6ngthdig(~)=O, I x1- I~I 1- dod6 11 2-n ( X ) lim w(x)= lim x g ~ =0, x -+ ~ x -+~ Ixl- SllY ra w lien t~ICt~li~. Do d6 w lien t~ICtren B. NI1lev~Yw th6acaeHnhcha'tHell trangdinh Iy. -Ta Cl)w=Klgl nen g=Klwl(theo dinh Iy llI.5-chlWng 4) ,SHYfa ll(X) :=:KI wJ(x) +hex) 11 2-11 I 1 112 1 =x w \ +CJ.) II x - n f(~)dS(~) Ixl aBlx-~1 '\JXE RII\ B, 0 .Yi~'~ht'l(i~l(am (jac 200/ 53 .-,A:f~~!lJJb.9h{UJt~% 3) Vi d{l: Xcl bAi loan Dirichlel d6i vdi mi€n ngoai ciia dla troll don vi B trangu? ') - L\u(x)=0, '\Ix E R-\ B u(x) =rex),'\Ix E 8B (i) (ii) lim HeX)==L IXI-7oo (hfj'uh~ln) (iii) lrungdt')r lien [~ICtrenbien8B vanghi~mllEC2(U?\B )nC(R\B). tHy changrninhkefqll(1sall: -Tnrongh(}pL=21Itff(l;)LlS(l;;): an Nghi~mcllabftito<1nlaLluynha'tvacobi611thuctrenR2\ B lEi ll(X) ==~ flxl2-I f(~)dS(~) 2ItaBlx- ~12 -Tn()ng IH}PL:;t:-f- ff(~)JS(O_It an BAi loan lren vo nghi~l1l. Cllllllg mill" Gia sLfbai loan co nghi~mla u. Th~oc1jl1hIy IV.2-chlfc1ng4 ,nghi~mu cod<;lng u(x)=w(x*)+h(x) v(1ih(x)= ~ flxl2_12It I x - r l 2 f(~)dS(~) an '-:> rex) lIeu ? - xER-\B neu x E8B ') Suy ra w(x*)=u(x)-h(x) , '\Ix E R-\B. Dod6 w(x)=u(x*)-h(x*) , '\Ix E B\{O}. Khi clH>X-7() trong B thl X*-7oo , u(x*) -7L (gi~lthief) va .(:I<;~ldL'f(?~{'(,;~o~c 200/ 54 ,jJ:j; 'j/b/L '%V/btf % h(x"') ~.~ ff(~)us(~)211: aB Do lit>khi x~O trungB thl (tinh chfttclia cong thltcPoisson) 1 f ' w(x)-->I.J -- l(~)dS(C;;) 211: au (ta gqi gidi h',111nay 1£1L' ) -SaudaytasedllYngminhw bang0 trenB\{O}, Xct hamg,lx)=w(x)+E(-lnlx ) IHlIl1gt;ui~uboatrenB\{O}. Khi X~Olhlgix) ~oo,suyra g,,(x»Onellx (higanO. l-)~tm=inf'{g,,(x):xEB\{O}}(mc6 thi la -.00). T6n t~liday{adlrongB\{O}saocho gt;(ak)~mkhik~oo. Day {adHamtrongt~pcompaGtB nent6nt<;1idaycon{bdhQi tu v6hEB , Neu hEB\{Ollhlhamdi~uhoag"d(,ltgia tr!clfetiiu khi x=b, lhcn nguyenIy Cl,retieu lhl g" 1£1ham hang (luon bang m) tren B\{0I,ma khix gan0 thlg,,(x»Onensuyra m>O.NelibE aB thlm=O dngo;{h)=O.Neuh=OthlIII ;?0 dog,,(x»Okhi x gaBO.Dod6 taluonc6 III ;:::O,suyra g,,(x)khongam trenB\{O}vdi mQiE >0,do d6 w(x) khtHlg[1mtrenB\{OI, Trongbiiu lluk cuagix) LaxcLhamdi~lIboa-w thayVIw,ly lu~11lu'dngH,rla c6 ket qua -w(x) khongam trenB\{OLt(rc fa w(x) khol1gllLrdngLrenB\{OI, Ta ua ch(rngIllinhw(x) VITakhongamVITakhongdlrOngtren B\{O},l1cnC()thesuyraw(x)bang0 LrenB\{O}. Dn (It>vabi6uth((ccliau LrenU,z\B la u(x) =w(x*)+h(x) =hex) ') =~ jlxl- - ~ f(C;;)uS(C;;)211: . l x-c;; I - au -Ham s6 u c6 biill th(rcd Ln3nth~tslf la nghi~mclla bai loan nell u Ihl1ade linh chftL(i),(ii),(iii).Ta Lhfty1Ithda(i)va (ii),cho x~oo Lhl u(x) ~~ ff(~)dS(~)211: an (j.) ( ",' '/' c.~) oLU-{?-/tFrz,/t C)(tu en t?c !to{N 55 Jt:;~<;yb;?!M(Mt~% do (() II chi Ih6a lInh chelL(iii) nc\, vi, chi nc'u L==~ fr(~)dS(~)2n an V~y Lacc>keLqu~idn chungminh. 0 4) Vi d{l XeL b[1iL(HinDirichlcLdoi v(Hmi~nngoai clia dla Lrondrin vi B trong n? , ') '-,- ,[.,II(X) == 0, \Ix E U-\ B ,11(x)==rex),\Ix E 013 .11bi ch~nd vungvoCl,fC, ILtcIii : 3M>O,3[>0,\Ix E U2\13,1xl>1'~ IHeX)I <M lrongde)r licn LI.letrcnbicn013viighi~mUEC2(U2\B)nC(n?\B), Hayeh(fngminhnghi~mcliabiii loanJii ') I f'x'-- J,l1(X) ==--- ? l(~)dS(~)2IT Ix -~ I -an Cldtllgmillh Thco djnh 19JV.2-chlwng 4 , nghi~m U c6 d<:1ng U(X)==w(x*)+h(x) Suy ra I )0 dc) f ? ~ Ixl-- J f(~)dS(~) vdih(x)== 1 2 , "aLlx-~12 rex) ". w(x*)==u(x)-h(x), \Ix E nhB. w(x)==u(x*)-h(x*) , \Ix E B \{O}. nell ? - XEU-\B IH~11 x EoB Khi cho X-70 Lh. X*-7W, hie de) h(x*) -7 ~ fr(~)dS(~)(1lnhchatctiac6ngLhuePoisson),2n an (IcingIhlJi lu(x *) 1<M khi Ix*1dti /dn(x* E n?\B), 1£' .'" ( ' (k/~j{{~i/{ Yrt/t' '(tv JL~C j!{j() / 56 J1:;;ljIblt5YicWthtfi do d6 I w(x) I hi d}~nhdi M+I~ ff(~)dS(~)1 khi x gan 0, DB XCLham g,lx)=w(x) -/-1=;(-Inlxl) . Ham g"t,li6uboa Iren B\{O}, Khi x~() ,Ihlg,,(x)~CX),SHYra g,,(x»Ontll x dLigftnO. f)[,ilm=inf{g,,(x):xEB\{OI} (m c6 Iht3Ia-CX). 'I'(inL"li day{adLrong B\{Olsao cho g,,(ak)~m khi k~CX), Day {ad nam Imng I~pcompacL13Ben tan t'.liday con {hdhQi t~1v€ hEB. Ncli hEB\{O}lhl ham di€u hoa g~d<;llgia Iri Cl,fCtit3l1khi x=h ,tbeo nguyenIy cl,fCLieu Lhlg~la hamhang(luanbangm) lren B\{0},makhi x gin 0 Lhlg,,(x»O nen SHYra m >O,N6uhE 8B thlm=Odo g~(b)=O,Ne'ub=O Lhlm ;:::()do gjx»O khi x gan O.l)o dt) Laluan c6 m ;:::O,suyra g,,(x) khang al1llren H\{OI vdi mqi E>0,do d6 w(x) khang am lfen B\{O}, Trung hit3uIinte cLiag~(x)ta xet ham di€u boa -w thay VI w,ly lu~n ll(ling llf Ll c() kc'tL/u.i-w(x) khong am IrenB\{0},t(rcIii w(x) khong dl(Ong Ire n B \ (() I, 'I'a dfJ ch((ng minh w(x) vua khong am vua kh6ng dl(0ng tren B\{0!,ncnc()theSHYraw(x)bang0 trenB\{O}, T6m l'.li,nghi~11lc la bai loan Iren Iii dllY nhatva c6 bit3uthl'i'ctTen n2\B la u(x)=w(x*)+h(x) =hex) =~ f lxl2-1 2rc I x -( 1 2 f(~)ds(~) DB ":> 0 5) Vi d{l: Xcl hai [min Dirichkt d6i vdi l1li€n ngoai clla dla Iron dl1nvi B trang n2, ~ - ,lHl( x) =0, \Ix E n.-\B ,u(x) =rex),\Ix E 8B u(x) .. -- I I ~ 0 kill x~ CX) 111X ( /J ," i /J ,.ij';; ,La~i/t frl/t ()(to ,;1(9(; 5!(){i/ 57 II' - 6.- cy/,-./ 'jI-a'pc-'ll-..:7h£Mthj/ /;; trong U() r lien l~IClren bienaB va nghit%mUE C2(R2\B) n C (U2\B). IHy clurng l11inhnghi~111CUi.!bai loan la ') 1 J 1xl--] u(x)=- -2f(~)dS(~)211: I x - ~I3B Cluj thieh :Vi cl~l3 (Iu(x) I bi ch~nkhi Ix I kha 16n)la lru'ongh(.jpd~c bi~lCUi.!vi cll.14 (I u(x)Ico th6lien ra 00khi x.~ oo).Chungtoi v~ntdnh bay vi cl~\3 vll11116nphanbit%lr5 tnc()nghc;iPu bi ch~nva khongbi ch~n. CluIng lIlillh Gd Sl~btli locinco nhgit%1111a1I. Thel>dinh Iy IV .2-chlcung4 , nghit%ml\ c6 d<;tng u(x)=w(x*)+h(x) vdi h(x)= ~ J lx12_1 211:3Blx- ~12-r(~)dS(~) rex) nell xER2\B nell x EaB Suy ra Do c1() ? w(x*)=u(x)-h(x) , \:Ix E R-\B. w(x)=lI(x*)-h(x*) , "dx E B\{O}. Khi chu x~O thl x*~oo, luc Ll6 h(x*) ~ J..- J r(~)dS(~)211: 3B (tinhcha'tcua congth(tcPoisson), Xet hamgJx)= w(x)+£(-lnlxl) ,vdi XE B\{O}. Ham gt;di~lIhoa trenB\{O}. Ti.! c6 gE(x) =u~x*)- h(x*) +Einlx *1 ( lI(X*) - h(x*) J u =: +£ Inlx -I-I Inlx *1 Khi X~O thl I x* I ~oo, gE(X)~oo,SHYra gt;(x»Onell x du gfinO. D~t l1l=inf{gt;(x) :xEB\{O}} (m c6 th61a -00). T6n l<.liJay{adtrong B\{O}sao cho gt;(ak)~m khi k~oo. (£' <'.' " ':/#' j{a}lt Frt-It 6au {/'['!c 200/ ss JfJt~t;jfblt<91{t;ltkc;{i Day {ad nam lmng l~pcompact B nen l6n l~i day con {bk}hQil~1v€ hE B. Ne'u hEB\{O}lhl h~lll1di~u hoa gt;d',ll gill If! Cl,tCtitSukhi x=b ,tIleD nguyen 19clfc li6u Lhlgt;la ham hang (Iuon bang 111)LrenB\{0},makhi x gan () Ihl gjx»O nen suy ra m >O,N6ubE aB thl m=Odo gt;(b)=0,N6ub=O Ihl m 20 do gt:(x»Okhi x gfin O,Dod6 ta luon c6 1112 O,suyra gE(X)khong am LrenB\{O}vdi mqi E>0,do d6 w(x) khong am LrenB\{O}, Twng hi6u LhC(clla g,,(x)la xet h~lIndi€u boa -w thay VI w,ly lu~n Il(dng It,rla Cl) kCIqua-w(x) khongam trenI3\{O},tC(cIii w(x) khongdl(Ong Lren Ii \ {° I. Ta da cIlll'ng minh w(x) vua khong am vua khong dlWng tren B\{O},ncnc()Ih6SHYraw(x)bang0 lrenB\{O}. Tl)l111',li,nghi~mdla bili LoantrenIii duy nha'tva c6 bitSutllll'Clren n?\B Iii u(x)=w(x*)+h(x) =hex) ? =~ f'X'- -~r(QdS(~)2IT I x-~ I - aB 0 6) \Ii d{l: XCI hili loan Dirichlet l!()i vdi mi€n ngoiii clla dJa Lronddn vi B trong, n- . ")- ,L\u(x)=0, '\IxE R-\ B ,u(x)=rex),'\IxE aB (i) (ii) u(x) ( kl ' ( "' ). _ II ~ L:;t) 11x~ CD III III X Lrungdo r lien ll,lCtren bien aB , L c6 thtSbang+CDhay -CD, Vtl nghi0m UE C2 (R2\ B) n C (Ie\B), Hay chCrngminhke'lquLlsaIl : . Khi L huntH~nIh1nghi~mIi}<.Iuynha'tva c6bi€u Ihuctrenn?\B la I flxl2- JII(X)=L .Inlxl+- -~r(c;,)ds(c;,)2IT Ix - c;,1 - DB .L(;ri/l 'l(z~( 0(;0 ,~(; ':;(J(J/ S9 A/' ~. ()// cp;' 'j/(t;7C/t ,j !z{{//tltj/ it . Khi L h~ng +0':) hay-UJ Lhlbai tOLlnvanghi~m, CIl {tllgIllillh Gj,i sil' h~\iLoan lren c(}nghi~m Iii lI. Theo dinh 19IY,2-chLwng4 , nghi~mII c6 ui:,lng lI(X)=w(x*)+h(x) Suy ra Do d6 [ ? ~- Ix/-~ f(t;;)uS(t;;) vl)i h(x)= 1 2n f i x _t;;1 2 DB rex) ') w(x*)=lI(x)-h(x) , '\Ix E n.-\B. w(x)=u(x*)-h(x*) ,'\IXE B\{O}, n~u xER2\B n~u X E 813 Khi cho x-}O thl x*-}O':), u(x*) lien wi +0':)hay -0':), h(x*) -} ~ ff(~)JS(t;;)2n DB llic li() w(x) lien Ldi +0':)hay -0':) , (Hnhchill clla cang LhucPoisson), Nhu'the ,l6n l,.li1'>0saoclIo w lllan Iuan dL((Jnghay Illan Iuan am tren ! mi~n13(0,1').M:;ttklulc,hiim s6 di~uhoa w tri~tLieutren bienClB. Theo dinh 19Il.3-chlcdng4 ,hamWc6 bi€ll thuetren B\{O}Hi w(x)=k(-Inlxl). '\IxeB\{O} Ll1c <-16u c()bi€u Lhtre[renn?\ B Iii . u(x)=w(x*)+h(x) =k(-lnlx*I)+h(x) =k.lnlx/+h(x) H~lIl1S()u c(}bieu thlk d [ren Lh~tsIr la nghi~mclla bai LOannell u lh6a dc linh chilL(i),(ii),(iii),Ta thily u lh6a (i)va (ii),cho X-}O':)till u(x) -}k Inlx! do (I()u lh()aLinhchill (iii) n~uva chi neu L=k va L hull h<;ln .~L;(i/' frt~1 (>{~Oi;t~(; !!()()/ 60 .A:f;t;YIJ/t!JZ,Wtli% V~y khi L hang +lfJ hay -0) lhi hai lOLinv() nghi~lll,con khi L hecnh<;ln lhi nghieIII u la tllIY nhal va co hi€lI lluk trenc()bi€u lh((cIren U?\B la ') I f ix1- - j u(x)= L .Inlxl+-- ') f(~)ds(~) 2n . I x-c-;; I ~ DB 0 7) Vi d{l: XCI hai loan Dirichlel dC)ivdi mi~nngoai cila quit cau d(Jn vi B lrang R" (V(1in>2) L1U(x)=0, '\IxE J{I\B (i) lI(X)=rex),'\IxE oB ( ii) lim HeX)=L (huu h<;ln)(iii) Ixl--+0) lrong (It)r lien Il.IClren bien oB va nghi~mUE C\}t'\B) n C (RlI\B). Hay chll'ngminhnghi4111cila bai loan la tllIY nhatva co bi€u lhuc tren }t"\ 13 Illll( S311 [ ) ') I 7-11 1 x ~-1 II(X)~ L - (0 ff(Ob(~) (1-lxl- )+(0 flx'- n r(~)dS(~) aB OBI c-;;I CIUl/lg Illillh GiJ sil'h~liloan c()nghi~mIii u. Thco tlinh Iy IV-I ,nghi~lllll c()tI<;lng u(x)= Ixr~-llw(x*)+h(x) r ~ f lxl2 - I . v,1i h(x)~1 (() aBlx- 1;,1"I(1;,)dS(1;,) lrex) nell X E nil \ B nell x EoB Suy ra w(x*)= Ixlll-2 (u(x)-h(x» , '\Ix E R"\B Do J() w(x)= Ixlz-n (lI(x*)-h(x*» , '\Ix E B\{O} :£:~i/t:Yrl~1(,(~OJt;(; !!!{){j/ 61 j r r?57/ CY/"" J(P1/blt J,1cvllh I'll- . Tnt'(Jnghl}PL=~ ff(~)dS(~) an XcI h{1l1lg,,(X)=W(X)+E( IxI2-Jl-l ) !-H1111g;:dicuhoa Lren13\{O}. Ta c() go:(x) ==IxI2-1I11I(X*) - h(x*)I+E(lxI2-n-1) ==(II(x*)-h(x*)+E)lxI2-II-E Khi cho x~O thl x*~C(), ILk d6 u(x*)~L va I h(x*)~ - ff(~)ds(~) coas Do lit) khi x~o tIll gAx) ~co, SHYra ge(X»OHe'llx lIll gan O. I-)~l111=inJ'{gAx) :xEB\{O}}(m c6 the la -co). (tinhchit cllatong thucPoisson) Tc1nL<.liday{adLrongB\{O}saotho g,,(ak)~mkhi k~co. Day {ad n~mLrungL~pcompact B uen L6nt~iday can {bdhQi ll~ve bE B. Nell hEB\{O}lhl h::l111(lieu hoa g" d~llghi Lr!cllc Lil5ukhi x=b ,theo nguyen Iy CIt'CLieu Lhlg" la ham h~ng(luau bang m) tren B\{O},makhi x gfin 0 thl gjx»O Hen SHYra m >O.Ne'ubE DB thl m=Odo gib)=O .Ne'ub=O thl m 2::0 do g,:Cx»Okhi x gftnO.Do d6 Laluon c6 m 2::O,suyra ge(x)khong am LrenB\{O) vdi nll.>iE>0,do d6 w(x) khongam tren B\{O}. Trong hi<5uLh(t'clla ge(x)ta XCI ham dicu boa -w thay VI w,ly lu~n tLt'dngLL.I'Lac6 ke'tquit -w(x) khong am trenB\{0},t((cIa w(x) khong dlt'dng lrcnB\{O}. Ta (la ch(rngminh w(x) vi'rakh6ng am vaa kh6ng dlWng tren B\{O},nenC()theSHYraw(x)bang0 trenB\{O}. Biell Ih(t'cllanghiC;111II lrenl{'\B la I I ')-il II(X) =X - w(x*) +hex) =hex) ') =~f'xl- -) f(~)dS(~) OJaBlx- ~III -H{lIll sf) u c6 bieu th(t'cd tn~nth~tSl.t'la nghi~mclla hai loan ne'liu lh6a dc Linhch[lt(i),(ii),(iii).Ta thiy u thoa(i)va (ii),cho x~co Lhl (jJ (,,' £' c:;w.' ,j{U,ilt Yrz,11 (5«(> (;7/((;(; Z(}(J! 62 Jf~u:!lb/?5lirvlz4r1i u(x)-+~ frU;')dS(~) DB do(16u Ih6aIlnllchat(iii) VIdangxellnfongh<;lp L=~ It'(~)dS(~) DB . Tn((jng h(Jp Lot:L fr(~)dS(~): DB Khi x-+() lhl +aJ neuL >! ff(~)dS(~)(0 DB w(x) -+ neu L <~ff(~)dS(~) DB Nhl!'Ih6 ,[(int0suocho w luon Juon dlwng huy Juan 1LJ()nam l1'l3n mi0n B(O,r), M~l khac, ham so di0u boa w lri~l lieu lren bien DB, Thco d!nh Iy II.3-chl!'dng4 ,hamw c6 biGuLInk lren B \{O}la W(X)=k(lxI2-1l-I) , \ixEB\{O} -UJ Loc dt) II co biGullul'cLIenu.11\B la ll(X)= Ix\2-1lw(x*)+h(x) =lxI2-1l.k~x*12-11- )+h(X) =k(I -lxI2-11) +h(x) [J H~lIn so u c6 bit311lh(fc d lren lh~l Sl,rIii nglli~m cila biii loan ne'll II Lh6acac llnll chat (i),(ii),(iii).Ta lhay II lhoa (i)va (ii),cho X-+aJ IhI ll(X) -+k+~ fr(~)dS(~) ,(0 DB do (I()II llH')allnh chat(iii) neu va chi neu Lo'c I;l L=k+~ fr(~)dS(~)CD DB k=L-L fr(~)dS(~) DB CI ) ,I ' ' 0 ';:(/)J {{(tit YrZ/t '( )(tu J'L rc !!(J()/ 63 j/' - 67/ OJ/.- J IftiJIblb <;/fliNt/if j/ (i V~y ngllicl1lu c() bi6u Iherc u(x)= [ L - ~ ff(~)JS(~)) (1-lxI2-")+~flxl:-,: f(~)ds(i;)DB DBIx ~I Ki/t [Will: G0p hai kct qua doi v(jj hai tHronghC,5Pclia L,ta Olrcjckit qui!dn chlfng minh, 0 8) Vid{l: Xct hui 10,lnDirichletdoi vdi mi~nngoaiclia qucldu odnvi B trung RII (vdi n>2) L\u(X)=0, "I/xE R'\ B HeX)= rex), "I/xE as (i) (ii) (iii)1i1l1 u(x)=L Ixl~oo lrungdo L bang+00hay -00, r lien tt,lCtren bien aB va nghi~rn 1 -- UEC-(Jt"\ 13)n C (If'\B), HayclufngminhhaiLOanIrenvonghj~rn. C'dlngminlt Giii sli'h~lito,1nc6 nghj~mlau, Thco d!nhly IV-l ,nghi~mucodi;lng 11 "-11 u(x)= x - w(x *)+h(x) SHY ra w(x*)= Ixl"-2 (u(x)-h(x» , "I/xE R'\ B Do lit) w(x)= IxI2-11(u(x*)-h(x*» , "I/xE B\{O} Khi x~() thl w(x)ticn toj +00hay -00, 1 pxl'-I . v(ji h(x)=i (D I -I" l(QdS() , neu x E R" \ B DBx rex) ' neu XEaS (~) ",. /) (jjt~ .'JcU;/dt Frldt' (j(?u (f[(;m 200/ 64 cA;~v;;g,? '%?/lZh cy;; NIH!'lhC~,ldn l'-;lir>Osao cho w lu6n Juan dlfcjnghay lu6n lu6n am tren mien B(O,r), M~t klHlc, h~\lns6 dieu hoa w tri~t lieu tren bien DB. Thco d!nh Iy I1.3-dn((jng4 ,hamw c6bi€u thliclrenB\{O}la II?-uw(x)=k( x - -I), \ixEB\{O} Luc do LIcobiEuthu'ctrenIf\B la II?-nu(x)= x - w(x*)+h(x) =Ix12-11.k~x*12-11- 1)+h(X) =k(I-lxI2-U) +h(x) Cho x~co lhl u(x)~k+~ ff(~)ds(~) 2nDB (lieunaymallthu£lnvdi Linhchat(iii). (huu h<.lIl), V~y h~liloan v6 nghi~m. 0