tổng quan về phương trình laplace

trình bày về tổng quan về phương trình laplace

I{t~ilt j/ (bit' ( ;(t(> ,f[ ~c ~O(J/ ,A;j;ryb/l 37z(///t~ 0;;:;

I M()T s6 DINH NGHiA

1 f)illh Ilghfa

- XCl n c RII va h~lIll so u: Q ~ R lhllQc hip c2 (D),

Toan lli' Laplacl; t<ICdl,\11gh~n ham so u (llrqc d!nh nghla bi~i

II

L1:= ~Dk

k=1 lrung cl6 I)~ la cl<,lOham rieng cftp hai,

- 1-)<.10ham rieng co nhi€u ky hi~lI kl1<lc nhall, do d6 L1u c6 th~ dl1"c;Jc

Vl~i x = (XI , " XII)E D,

II

i) L~U (x) = IDk~I(X)

k=1

II) L1U (x) = I ~lI(X) = I ~U(Xl""'XII)

II

iii) L1U(x) =LlIXkXk (x)

k=l

2) nillh Ilgltfu

I-Hu11s(i u dLrl}cgQi la ham di€u boa tren D neu

CIIiI/hie"

. Khi dinh nghla h~un di€u boa, ngLreji ta co lh~ xel h~lI11II c6 gia tr!

pink, lilY nhien lrong lu~n van nay cluing loi chI xer h~lIllsO'u c6 giJ lr! Ll1Lrc,

5t«ldt (f;~~(' f)(~,/ L~C 200/ 2 t/" - 1:'77/ OJ/,:

Trong lu~n van nay,chung loi c~li XCI mi~n xac d!nh clla ham aieu

hoa Iii qp con clla R" vdi n ~ 2,

. Thay VI ghi nhu' (1), ngLt'oi la c6 lh~ ghi la :

"~u bhng 0 lren n"

3 lJillll Ilgllia

- Phlfdng lrlnh Laplace la phlrdng lrlnh co lh,lng

~u (x) =0, vx E n trong lit) n c R" va u la ham s6 din 11m,

- Bil i lOan Dirichlet d6i vdi phlfljng tdnh Laplace Iii bai loan co 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 lrong dt) r Iii ham S6l1llrc lien H,IClIen an,

- Hili loan Neumann d6i vdi phl(dng ldnh Laplace la bai lOan co di;lng nhu' sau :

Tim u E C2 (D) n C (D) lh6a

l

~u(X) = 0, vx EQ ,

au

au

lrung J6 g Iii ham s6 IIWc lien t~IClren aD va ~~ (x) la d"to ham clla u ti;lix

Ihcn hlt'dng clia vecW u , vdi u la philp vecW don V!hlt'dng ngoai clia bien

an,

4) Cfllt tllicll

Phlt'dng lrlnh Laplace IllY la plHfl1ng trlnh d~1Oham rieng cd b,ln nhlt'ng ra't quan lH,>ng VI n6 xua't hi~n nhi€u lrong v~t Iy, chang h~ln trang tHrong llnh Ji~n, lH,Jng lnt'ong ,

- Xc [D Iii mQtlllien md, bj ch~n, lien [hong co dura IHrong LInhdi~n,

J-)i9n Ihong qua m~t av bhng ()

~0;(ill }(t~t (Itt A'~c 200/ 3 L/tj;t;1'8n 5n'cvllh %

av

lrong tit) E IiI vecW ClrolIg dC?di~n wrong va u la phap vecltf dOn vi htrdng

n6 11

TheD d!nh 19 Gauss - Green lhl

v

))0 th) divE =0 Iren n

Vdi II IiI di~1I the' lhl E =- 'Vu

SllY ra div (Y'u) =0

Y~Y 0.11=() Ircn Q

- Trong cac tHrong khac, vi~c thie't l~p pln((jog trtnh Laplace tt((jog tt.!' nhlr trung In'-ling nnh di<:;n

J) nillll l.y:

XcI hili 10<ln ll\ll1l E C2 (R'\ {OJ) th6a

f L1u(x)=O, VxERn\{O)

l

-::;- + C, o:?:-

Ixlll lrung (II) h,c lit cac hang so

CIl "ollg millil

Ch(fng l11inh clla dinh 19 nay dtrdc lrlnh hay d lrang 21 cuon sach [11

2) nillll Ii:

B(O,r)= Ix E l{": x 1< r}

I ({(.bIt f/ /Z II {'((D J[ 9G!!OO I 4 ~Ajl(;jlg'/t 5fZtt'/tjz %

{

L'lll(X)= 0, Vx E 13(0,r)

HeX) = rex), Vx E 8B(0,r)

lrong lh) ria ha m lien ll,c L1'enbien aB

Nghiqm lIEC\B)nC(B) clla bai loan Hl LIllYnhal va Cl) bi6u lht(c lren B(O,1') nluf sall

') 11

2

lICK)=- f f(~)dS(~) , Vx E B(O,r)

(Dr

I I

u

lrong d6 (t) IA di~n lich clla m~Hcall odn vi 813,

C{)ng Ihu'c clla 1I d lfen ciL(ejcgqi la c!lfC.kgqi li\ c6ng lhtfe Heh phan

Poisson, nell dall'J(~,x)= ~ r x Ihl c6ng Lhtfc(ren LrdIhanh

mr I~-xlll

lICK)= f H(~,x)r(~)dS(~)

1~I=r

(,/ut Ilticlt: Trong lrt(dng helP n=2 lhl II c6 IhtS Viellheo 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)

Chu'ng minh cLia u!nh 19 lren du'cjc Irlnh bay d lrang JO7, quyGn sach 121

Clllt tlticlt:

Tinh chflL clia 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, khi 06 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?th c;{i

{~E /(11: I~I =l' va I~ - ~I> 0 > O},vdi 0 Hiso dl(ongtoy y,

~ ~', ~,." ;:: ,

III MOT SO TINH CHAT CUA HAM DIEU HOA.

1 Dfllit l.v(cong tlnk gici trj trung binh) :

Di~u kil$n dn va au d6 II Iii hiim so' di~ll hoa tren £1Ia

HeX)= ~I-I fu(~)dS~, VB (x, 1') C £1

cur

3B(x,r)

trung d6 U)Iii di~n rich clia m~t dll dcln vi va C0l,n-1la di~n rich clia m~t du ban kinh L

Cflli thieh:

Ne'u linh trung blnh rhea rich phan kh6i thl ham oi~u hoa v~n thoa, tac

Iii

ll(X) = ~ fu(y)dy

Vel') l3(x,r) trung (\() V (r) ILlth6 tich clia CIliadu ban kinh r .

Cflli'llg millh

Binh Iy tren Ol(QCchang minh a trang 25, sach [1]

2 Hillh if' (nguyen l.vqtc ([{Ii):

ch~n R").

a) Khi (\()

max u = max u

b) NcLin Iii l~p lien thong va t6n t~lix" E Q saG cho

u(x,,) = nE~xu

Q

Khi d() II la hiu11hang tren £1,

Chli'llg millh

:£~~l;t ')/(Z~('6r~o (7f~c 200/ 6

L/):?':!Ie'7t~ c/ /l{!//lh f/ it,

Dinh Iy Lren cfL(c.Jcch(tng minh d trang 27, sach [I]

3, Vi/lh ij (nguyen ly q(C liiu) :

Gi~1sli' [I E Cl(D.) n c (D.)

neR",

LI(Xo)= nun u ,

n

khi de) LI lil hill11 hang tren n,

4 j-Ji/lh iy (djnh ly Harnack) :

Giii sLl'n lil L~p md lien Lhong Lrong Rn, K la t~p compact nam trong n, Khi dt) tl)n Lai hang so c E (0,1) sao cho,

LI(X) I cS;-S,-Hey) c

Xcm trang 33, sach [I]

5 fJi/lh i5'(nguyen ly Harnack) :

boa Lang Ll'fngditSm Lren D.

Khi dt), mollrung hai lnWng hCJpsau xay ra

a) VXEn, LIllI(X) -+ + (f.)khi m -+ 00,

h) '1\111Lai1110tham s611 cli611boa lren n sao cho lulu} h0i tl,ld6u v6 u

tren mqi L0P compact Ken,

UaL VII'= [l1I1-[ll+llhl vwdlWng vt1i mqi m

a)CJiii sLi'lAn t(,li XEn lh6a LI1I\(X)-++00khi m-++oo:

.Z ;~i/t '1ft ~t '(ftO (;(~c !lOOI 7 J'f;1~cYiJ~t 5flw/l1Z c;:;

Cui Y LllY9 LhllQc Q

T~p K={x,y} Iii t~p compacL tmng Q

1\p dl,lng djnh 19 Harnack(djnh 19 1I1.4-chlrdng I) dol vdi K ,t6n t~i hang s6 C E( 1,0) Lhoa

"dI11E N , 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 *CIJLhi vii tnli Lien ra vo clfc ,do d6 um(Y) *CIJ

h)Gi,1 sll' lim llllJ(x) t6n t<.lihall h~ln vdi I11qiXEQ :

111~ ex)

1)~L u(x)= lim Ulll (x)

m~oo

Coi K I~lL~p compact toy ylrang Q

c6 dinh xEK.

1'6n L<.lihang so cE(1,CIJ) thoa bat d~ng lh((c Harnack(djnh 19 111.4-chl(l1ng I)

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 *(/.) ,La Ul(}C

] IU(Y)-llj(y)1 ::; -lll(X)-Uj(x)1

C

1)0 dt')

c

=> IlI(Y)-Uj(Y) < £

V~Y U lien LlJCJ~u Lren K

0

6 Dj/llllj:

~Yi~(t/t 'f;t~( '(~(~O,/f;,c 200/ 8 A;~t;!Iblt $CVltA: ni

Khi lit>

a) 1I I~Ih~lll1 giiii lieh lren O

h) Vdi a lOYYlhllQC 0, l6n l~i mQl Ian c~n Va clia a sao cho chlloi

Toylor clla 1Ih()i ll.1llly~l d6i va d6u lrong Ian c~n Va.

Chlloi Taylor clia II l~liJan c~n clia a la

00 D(xlI(a)

I(XI=o .

hoac viet dieh khac nhl( sall

00

III=0

lrung lit') Pili(x - a) = LCo:.cx - a)(:(

lal=m c)1:)a llllk Pili trong khai tri€n clia ham di6u hoa u cling la ham di6u boa lren V"

Dinh Iy lren dU\5cch((ng minh IH5ilrang 31, sliGh III va trang 24 sliGh 13\.

IV TiNH IHJY NHAT CUA NGHII~M

1) lJj/l1t (v :

Giii Sll'0 la t~p md hi ch~n hong nil va fEC(aO)

gia lrj bien Dirichlet

{

.6.1I= 0 lre n 0

U = f lfen 00

CIUI/lg 11li/lh

Xcm lrang 28, s,'ich 11],

2 lJjlllt i}:

h;(illfrt~f (rt<, jit;(; J!OO/ 9 J(y-":jl6'lt ,I" L7lilV/lli v-v: OJ/,: f/ t'i

Giii sll' Q fa l~p md, bi ch~n, lien thong trong RI1va r E C(Q),

Xet nghiQl11u E C2(Q) aD'i voi bai loan gia lri bien Neumann

{

du

J' ~

an.

du

h~lIl1hang,

C/Hlllg lIlillh

Vdi lI, v E C2( Q), lheo c6ng lh((c Green la c6

XC! v = II va u la ham di~u h6t\ (ham c6 ~u ::: 0) lhl cong thac tren trd

lha nl!

fL

f

du

Xi du

Ncli f=() lhl ~ lriet lieu bien aD ,do d6

II

ni=l

lLi'c 1[11I 1[\ ham hang tren 0.,

Gia sLYhai lOan Neuman c6 hai nghi~m u\, U2 E (D),

h' I ' I,'" I ' C ?(n.) " dll ~ b'~

I-)al 1I::: lI, - lh t 1 1(\)11(leu 10a u E -~.! va co -::: 0 lren ten

aD.

Thcn kC'll1l1ii clia bl(OC 1 lhlula ham hang lren D,

0

,~L~t;t 'f(l~{ '0(~(, i/i'~D 20[J! 10 ,A;~,,?g1t 3,icMIZ %

('1111thich: Xet vdi bai loan ghl trj bien Dirichlet dO'i vdi mi<snngoai

clla 0 (vdi 0 la t~p md, lien th6ng, bi ch~n trong R")

{

llU = 0 u=f

tren tren

1(11\0

an

Bai Loan tren c() the c6 nhi~l1 nghi~lp Chung toi trlnh bay chi tie't d chl((jng 4

V TINH TON Tl}I CUA NGHII}:M:

- Xc Lba i to,ln Dirichlet tren mi<snn

fllu =0 Lren n

(llicn ltlc lren bien an)

- Trung Lll((Jng IH.jpn la qua call B(O, r) thl bEd Loan LIen c6 nghi~m Lluynhfll va hi611Lhacciia nghi~m da dl((JCtrinh bay trong dint 1911.2.

dura chac c() nghi~m l:)i~u ki~n v~ n de bai Loan c() nghi~m se dl(CJCtrinh bay chi tiel Lrung d1l(Ong 3