Một số tính chất của hàm điều hòa

x =(XI ,.." XII)E D, II i) L~U(x) =IDk~I(X) k=1 ,. II a'2 II a'2 II) L1U(x) =I ~lI(X) =I ~U(Xl""'XII) k=laxk k=laxk II iii) L1U(x) =LlIXkXk (x) k=l 2) nillh Ilgltfu .. XCl Q c H."vahamsoLlll.rCu E c2 (D). I-Hu11s(iudLrl}cgQilahamdi€u boatrenD neu L1II(X)=0, '\Ix E Q (1) CIIiI/hie" .. . Khi dinh nghla h~undi€u boa, ngLrejita co lh~ xel h~lI11II c6 gia tr! pink, lilY nhien lrong lu~nvan nay cluing loi chI xer h~lIllsO'u c6 giJ lr! Ll1Lrc, 5t«ldt(f;~~(' f)(~,/ L~C 200/ 2 --t/" - 1:'77/ OJ/,: Jf;pvye'/t JhiV'/th 1"/; . Trong lu~nvan nay,chungloi c~liXCI mi~nxac d!nhclla hamaieu hoaIii qp conclla R" vdin ~2, . Thay VI ghi nhu'(1), ngLt'oila c6 lh~ghi la : "~ubhng0 lrenn" 3. lJillll Ilgllia .. - Phlfdng lrlnh Laplace la phlrdnglrlnh co lh,lng ~u(x) =0, vx E n tronglit)n c R" vau lahams6din 11m, - Bili lOanDirichletd6i vdi phlfljngtdnhLaplaceIii bai loanco d~lng nillt'sau: Tim u E C2(n) n C (n) lhoa f~u(x) =0, vx En 1 u(x) =rex),vx Ean lrongdt)r Iii hamS6l1llrclienH,IClIen an, - Hili loanNeumannd6i vdi phl(dngldnhLaplacela bai lOanco di;lng nhu'sau: Tim u E C2(D) n C (D) lh6a l ~u(X) =0, vx E Q , au -(x) =g(x),VxEaDau lrungJ6 g Iii hams6 IIWclien t~IClrenaD va ~~(x) la d"tohamclla u ti;lix Ihcnhlt'dngclia vecWu , vdi u la philpvecWdonV!hlt'dngngoaiclia bien an, 4) Cfllt tllicll .. Phlt'dnglrlnh Laplace IllY la plHfl1ngtrlnh d~1Oham rieng cd b,ln nhlt'ng ra'tquan lH,>ngVI n6 xua'thi~n nhi€u lrong v~t Iy, chang h~lntrang tHrong llnh Ji~n, lH,Jnglnt'ong.., - Xc [ D Iii mQtlllienmd,bjch~n,lien [hongcoduraIHrongLInhdi~n, Coi v la I mienIronnhmtmngn J-)i9nIhongquam~tav bhng() ~0;(ill }(t~t (Itt. A'~c 200/ 3 L/tj;t;1'8n5n'cvllh% f Ku dS=0 av lrongtit)E IiI vecWClrolIgdC?di~nwrongva u la phapvecltfdOnvi htrdng ngoai CllaI11ZiI av. n611 TheD d!nh19Gauss- Green lhl fdivE dx=f E.u dS v av fdivE dx=0 v ))0 th) divE =0 Iren n Vdi II IiI di~1Ithe'lhl E =- 'Vu SllY ra div (Y'u) =0 Y~Y .0.11=() Ircn Q - Trong cac tHrongkhac, vi~cthie'tl~ppln((jogtrtnhLaplace tt((jogtt.!' nhlr trungIn'-lingnnhdi<:;n. II. N(;HII!~!VITRONG MOT SOTRVONG H(iP D~C nl~T J) nillll l.y: XcI hili 10<lnll\ll1l E C2 (R'\ {OJ) th6a f L1u(x)=O, VxERn\{O) III (x) labitSuthaclheo I x I Nghi~111clia bAi loan co d<;lngoInt san l bInlxl+c, n ~ 2 u(x)= h 3---::;- +C, o:?:- Ixlll-- lrung (II) h,c lit cac hang so. CIl "ollgmillil Ch(fng l11inhclla dinh 19nay dtrdc lrlnh hay d lrang 21 cuon sach [11 2) nillll Ii: XcI h:li loan Dirichlel uoi B(O,r)= Ix E l{": I x 1<r} vdi mien ~ong clla qud cAlI (/' '//'. /' '-ff-' I ({(.bIt f/ /ZII {'((D J[ 9G!!OO I 4 ~Ajl(;jlg'/t5fZtt'/tjz% { L'lll(X)=0, Vx E 13(0,r) HeX)= rex), Vx E 8B(0,r) lrong lh) ria ham lien ll,c L1'enbien aB. Nghiqm lIEC\B)nC(B) clla bai loan Hl LIllYnhal va Cl) bi6u lht(clren B(O,1')nluf sall ') 11 2 I r-- x lICK)=- f f(~)dS(~) , Vx E B(O,r)(Dr I I u 1~I=r~- x lrong d6 (t)IA di~nlich clla m~Hcall odn vi 813, C{)ngIhu'cclla 1Id lfen ciL(ejcgqi la c!lfC.kgqi li\ c6ng lhtfe Hehphan ? 112 Poisson, nell dall'J(~,x)= ~r-- x Ihlc6ngLhtfc(renLrdIhanh mrI~-xlll lICK)= f H(~,x)r(~)dS(~) 1~I=r (,/ut Ilticlt: Trong lrt(dnghelPn=2lhl II c6 IhtSViellheo lqa 00 Cl,fCnht( sall I 211 ] - 1'2 lIeI', 0) = f i«p)d(p 2IT n 1'2+] - 2rcos(8 -tp) ,'V(r,8)E [0,1)x[O,2IT) C'dlng lIlinh Chu'ng minh cLia u!nh 19 lren du'cjc Irlnh bay d lrang JO7, quyGn sach 121 Clllt tlticlt: Tinh chflLclia H nlllr SilU (a) II(~, x) E COO (h) L'lJI (~,x) =0 (c) fH(~,X)dS~= I 1~I=r (LI)H(~,x»Oneu kl =r, Ix I <I' (c) Nell I ~I =r, khi06 lim H(~,x)=0 X-7~ Ixl<r neB kl~r, Ixl*r,~*x. Hell Ixl*r, I~I =r. neB I xl<1'. "Y;;rf/I 'Pr(~i(0(;0 Yt(;c 200/ 5 ejtj:;t;;yJn ,71:a?thc;{i vagi(iihi}nnayd~utheo~trenmi~n {~E /(11: I ~I =l'va I ~- ~I >0>O},vdi0Hisodl(ongtoyy, ~ ~', ~,." ;:: , III. MOT SO TINH CHAT CUA HAM DIEU HOA. 1.Dfllit l.v(congtlnkgicitrj trungbinh): Xct n Ii'!t~pmdtrongWIvauIa hamso'th~(cthuQcc2 (£1) Di~ukil$ndn vaaud6II Iii hiimso'di~llhoatren£1Ia HeX)= ~I-I fu(~)dS~,VB (x, 1')C £1cur 3B(x,r) . trungd6 U)Iii di~nrich clia m~tdll dclnvi va C0l,n-1la di~nrich clia m~tdu ban kinh L Cflli thieh: Ne'u linh trungblnh rhearich phan kh6i thl ham oi~uhoa v~nthoa, tac Iii ll(X) =~ fu(y)dyVel') l3(x,r) trung(\()V (r) ILlth6tichclia CIliadu bankinh r . Cflli'llg millh Binh Iy trenOl(QCchangminha trang25,sach[1] 2.Hillh if' (nguyenl.vqtc ([{Ii): Giii SL(1IE c2 (£1)n C (£1)la hamc1i~llboatfen£1(vdin Iii t~pmdhi ch~n R"). a) Khi (\() maxu =maxu Q , ail b) NcLin Iii l~plienthongvat6nt~lix" E Q saGcho u(x,,)= nE~xu Q Khi d() II lahiu11hangtren£1, Chli'llg millh :£~~l;t')/(Z~('6r~o(7f~c200/ 6 ,1/' ~, t7T/ c//.' L/):?':!Ie'7t~ c/ /l{!//lh f/ it, Dinh Iy LrencfL(c.Jcch(tngminhd trang27, sach [I] 3, Vi/lh ij (nguyenly q(Cliiu) : Gi~1sli'[I E Cl(D.)n c (D.) neR", Iii h~llnui~uhoa tren t~pmd bi ch~n a) Khi lit) minu=minu Q aD. h)Ne~LIn lien thongva tdnt<;liXl)E n saocho LI(Xo)= nunu , n khi de)LI lil hill11hang tren n, 4. j-Ji/lh iy (djnh ly Harnack) : Giii sLl'n lil L~pmd lien LhongLrongRn, K la t~pcompactnam trongn, Khi dt) tl)n Lai hangso c E (0,1) saocho, LI(X) . IcS;-S,- Hey) c Vt1imoi hill11di~uboadlwngUX,1Cdinhtrenn va vdi mqiX,y E K, Cllll/lg Illillh Xcm trang33, sach[I] 5. fJi/lh i5'(nguyenly Harnack): (.iii sli' n lil qp md lien thongtrungUn , {UlIJ la day cac hamui~u boaLangLl'fngditSmLrenD. Khi dt), mollrung hai lnWng hCJpsauxay ra a) VXEn, LIllI (X) -+ + (f.)khi m -+ 00, h) '1\111Lai1110thams611cli611boalrenn saocho lulu}h0i tl,ld6uv6u tren mqi L0PcompactKen, Cldi/lg Illi/lh UaL VII'= [l1I1-[ll+llhl vwdlWng vt1imqi m. a)CJiii sLi'lAn t(,liXEn lh6a LI1I\(X)-++00khi m-++oo: .Z ;~i/t'1ft~t'(ftO (;(~c !lOOI 7 J'f;1~cYiJ~t5flw/l1Zc;:; Cui Y LllY9 LhllQcQ. T~p K={x,y}Iii t~pcompacLtmngQ. 1\p dl,lngdjnh 19Harnack(djnh 191I1.4-chlrdngI) dol vdi K ,t6n t~i hang s6 C E( 1,0)Lhoa "dI11EN , C vlllx) < vlIlY) SlIY ra C(UlIlX)-lIl(X)+I) < ulI,(Y)-UI(Y)+1 C(uII1(x)-ul(x)+I)+lll(Y) -I <1l1ll(Y) Klli I1I *CIJLhivii tnli Lienra vo clfc ,dod6 um(Y) *CIJ. h)Gi,1sll' lim llllJ (x) t6n t<.lihall h~lnvdi I11qiXEQ : 111~ ex) 1)~Lu(x)= lim Ulll (x). m~oo Coi K I~lL~pcompacttoy y lrang Q . c6 dinhxEK. 1'6n L<.lihang so cE(1,CIJ) thoa bat d~ng lh((c Harnack(djnh 19 111.4- chl(l1ngI). Coi i la st{nguyendlWngtoy9. I "dYEK, "d111>1,(ll'll-llj)(Y)::; -(UII1-Uj)(X) c I IlIllI(y)-uj(y)1::;-IUIlI(X)-llj(x)1 c Chu m *(/.),LaUl(}C ] IU(Y)-llj(y)1 ::; -lll(X)-Uj(x)1 C 1)0 dt') "d£>O,3NEN ,"dYEK, "diEN, i >N => !IU(X)-Ui(X)1 < £ c =>IlI(Y)-Uj(Y) <£ V~Y U lien LlJCJ~u Lren K. 0 6. Dj/llllj: Giii sLi'n la L~p1l1dtrongUnvaUla hamsothl,fcdi~uhoaLrenQ. ~Yi~(t/t'f;t~('(~(~O,/f;,c .200/ 8 A;~t;!Iblt$CVltA:ni Khi lit> a) 1I I~Ih~lll1giiii lieh lren O. h) Vdi a lOY Y lhllQC0, l6n l~i mQl Ian c~nVa clia a sao cho chlloi Toylorclla 1Ih()ill.1llly~ld6ivad6ulrongIanc~nVa. Chlloi Taylorclia II l~liJanc~ncliaa la 00 D(xlI(a) HeX) ='" (x- a)a ~ a' I(XI=o . hoac viet dieh khac nhl(sall 00 u(x)=LPm(x-a) III=0 lrung lit') Pili(x - a) = LCo:.cx- a)(:( lal=m c)1:)allllk Pili trong khai tri€n clia ham di6u hoa u cling la ham di6u boa lren V". Cfdl/lg 11li/lh Dinh Iy lrendU\5ch((ngminhIH5ilrang31,sliGhIII va trang24 sliGh 13\. IV. TiNH IHJY NHAT CUA NGHII~M 1)lJj/l1t(v: Giii Sll'0 la t~pmdhi ch~nhongnil va fEC(aO) . Khi lI6 t6n t~litoj da mQtnghi~mII E C2(O)nc(o) d6i vdi hai loan gia lrj bienDirichlet { .6.1I=0 lren 0 U=f lfen 00 CIUI/lg 11li/lh Xcm lrang28,s,'ich11], 2. lJjlllt i}: h;(illfrt~f (rt<,jit;(; J!OO/ 9 ,I " v-v: OJ/,: J(y-":jl6'lt L7lilV/lli f/ t'i Giii sll' Q fa l~pmd, bi ch~n,lien thongtrongRI1va r E C(Q), Xet nghiQl11uE C2(Q) aD'ivoi bai loan gia lri bien Neumann { ~U= 0 lren Q du J' ~ an.- = lren .H. du (lmng ell) u la phap veclO adn vi hl(dng ngoai cllt\bienao.) Khi ell)caenghi~mthll()eC2(Q) (ntu t(int<;ti)sesaikhacvoi nhall Il1Qt h~lIl1hang, C/Hlllg lIlillh -- IhMc 1 (In(Ong h<.Jp1'=0): Vdi lI, v E C2(Q), lheo c6ng lh((cGreen la c6 n d Iv ~udx=- fL VXj UXjdx + fv d~dS n n i=l an XC! v =II vau la hamdi~uh6t\(hamc6 ~u::: 0) lhl congthactrentrd lhanl! fL lI ') f du 0=- u~dx+ u-dS Xi du n i=\ an Ncli f=() lhl ~ lrietlieu bienaD ,dod6 du ' II JLu~idx=() , ni=l lLi'c1[11I1[\ham hang tren 0., - IhMc 2 (In(OngIlljp rba'tky): Gia sLYhai lOanNeumanc6 hai nghi~mu\, U2E (D), h' I ' I ,'" I ' C ? (n.) " dll ~ b'~I-)al 1I::: lI, - lh t 1 1(\)11(leu 10au E -~.! vaco -::: 0 lren ten . - du aD. Thcn kC'll1l1iiclia bl(OC1 lhlula hamhang lren D, 0 ,~L~t;t 'f(l~{'0(~(,i/i'~D 20[J! 10 ,A;~,,?g1t3,icMIZ% ('1111thich:Xet vdi bai loanghl trj bienDirichletdO'ivdi mi<sngoai clla 0 (vdi 0 la t~pmd,lien th6ng,bi ch~ntrongR") { llU =0 u=f tren tren 1(11\0 an Bai Loantren c() the c6 nhi~l1nghi~lp.Chung toi trlnh bay chi tie'td chl((jng4. / A ., A V. TINH TON Tl}I CUA NGHII}:M: - XcLbai to,lnDirichlettrenmi<sn fllu =0 Lren n lu=r LTen an (llicn ltlclrenbienan) - Trung Lll((JngIH.jpn la quacall B(O,r) thl bEdLoanLIenc6 nghi~m Lluynhfllvahi611Lhac iianghi~mdadl((JCtrinhbaytrongdint1911.2. - Trung lH((Jnghc.jpn la t~pmd,bi ch~ntllY 9 Lrongl~Uthi bai Loan dura chac c() nghi~m.l:)i~uki~nv~n de bai Loanc() nghi~mse dl(CJCtrinh bay chi tiel Lrungd1l(Ong3. ._.