Luận án phó tiến sỹ Chỉnh hóa một số bài toán ngược trong khoa học ứng dụng

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Chuyennganh: loAN GJArItCH
\13 s6 01.01.01
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TOM T£(TLu.~N AN
Fh6 Tie'nSIKhoahQcToan Ly
Thanh ph6116 ChI Minh
II

- 1996-
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Lu~n ~n n1iydu'Qchean thanh t~i Khoa Toh - Tin h9c II
Tru'CJngBl}i hQc &boa hQc TV lJl'Mn Thanh pho' hI6 Chi Minh
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Lu~n ~n se du'<;fcbaa v~ tq,iH9i D6ng Chill Lu4n an Nh;) Nll'OChqp
tq,iTru'CJngDq.ihQcKhoa h9CTv Nhien Thanh pho' H6 Chi Minh VaG
hk~ giCJ ~ ngay thclng ~ Ham 1~96. '
III

III
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C6th! tlm hilu Lutjn dn tQi cdc tllltvifl1 : !:I
-' Tnto/'lg Dqi h9C Khan h9C T~(Nhien Thc'rl1hpluJ'H6 Chi Minh
- Khaa H9c nfng fir]) Thanh ph/f H6 C?llMinh
aID
nO GIAo D~JC vA.BAo L'}O
D/\I HQC Quc5c CIA THANH PH6 HO CHi MINH
TRU<JNCDAI HOC KHOA HQC TV NHlt N
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NGUYEN C(1NG TAM
cHiNn HOAj\U)Tso nAI ToAN NGU<;jC
TRONG KI-IO;\H()C (fNG D\JNG
Chuycn nganh: ToAN GIAr TfCH
Mil s6 : 01.01.01
, ,I .,

TONI TAT LUr'~N AN
~h6 Tien sl Khoa hqc Toan Ly
Thanh ph6 H6 Chi Minh
- 1996-
f'
Lu4n an n~y du<;1choan thanh t~i Khoa Toan - Tin 119C
Tnfong D<~ihQc Khw bQc Tlj N11ienThanh ph6H6 Chi Minh
Ng!-!QjJilltjn~ dKI! :
GS TS DANG DINH ANG
~-IDi(jUthill!-,TIiLLl
~gtiotllh;!11_'!fcL~_l
Cd (Juan Itlliin xet :
LUi)n illl se dlNc bao vi!:tqi HC)iDdng Cbi(m Luqn an Nhii Nltdc bqp
v,"iTntongD~i hqc Khoil hQc 1'11Nhien Thanh pIle)Ho Chi Minh yito
hie __gio_ngay___thilng_nam 1996,
Co thE lim hilu Lu4n c!ll'~Iiede Ilzz(vifill :
- Tn((/ng Dgi Iz~)c FI.hoa h'lc IV Nhien Thell/h 1711()'JJ(3 Chi Minh
- Klzoa H,?e Tllng Hq'p Thanh pMJ'H(5Chi Minh
MO 8AU
Trong Khoa hl)c ling d\Ing, nolI du khao sat hili loan nglic;1cdii xullt
hi~n tit' lau, Coo de'n nhung nam 60, d6ng thdi VOlvi~c phat tri6n cac c6ng C\l
loan hQC,cac hiii loan ngu'<1ckh6ng chino dii du'<1ccac nha loan hl)C tren the'
giOi khao sat m(Jt cach sau r(Jng ma lieu bi6u lil t:ac c6ng trinh clia Tikhonov,
Lavrentiev, Lions, Tit'thdi gian do de'n nay, cac bai loan ngu'<1ckh6ng chino
ngay cang du'<1cchu 9 khao sat m(Jt cach r(Jng riii do nhung nhu du xullt pilat
tit'th,,'c te' cua khoa hl)c ling d1!ng, d~c bic$ttrong Ky nghc$,Y hl)c, V~t Iy Oia
du,
Trong Lu~n an, chling Wi khao sat m(Jt s6 hili loan nglt<,1cco a~ng
trong do r lit au ki~n nh~n du'<,1c(qua quaIl sat, do d~c), h9 th6ng A la mOtphuUng
trinh d;;toham rieng VOlcac di~u kic$nbien tuong ling va v ill du kic$ndn tlm.

Trong lu~n an, chung t6i khao sat mQt s6 bili loan Cauchy coo phu'ong trinh
Poisson va Laplace trong doc mi~n khac nhau ct'Ia R2 va RJ. Nhil'ng bili loan nay
co 9nghia qua" tn,mg trong ling d\Ing, chAng h;;tnnhu' trong V~t 19Dja du, VI
nghi9m ctta cac hili loan nay se du'<,1cxac dinh khi ta hic't di~u kic$nDirichlet (hay
Neumann) tren loan bien mi~n khao sat nen vic$cgiai hili loan Cauchy coo
phuong trInh Poisson hay Laplace du'<,1ccoi nhu bili loan tlm du kic$nd~u vao v la
di6u ki9n hien Dirichlet (hay Neumann) khi bic't du kic$nd~u ra F la di0u ki~n
bien CalJ<.:h'y(trcn mQt ph~n bien) va hc$thi')ng A chino la phut1ng trInh Poisson
hay Laplace tuUng (tng, hong D,a V~( Iy, hili loan nilYc6 9 nghia th\l'cto vlugu'\fi
ta thuong khang (ht; do d;;tc gia tri tntong trl)ng lifc, tr9ng 1\l'caj thll'ong hay
gradient dw n6 tren loan bien m't chi c6 th6 do tren mOtph~n bien ma thai,
- I -
!-)u vilO
I
H th6ng
!-)u ra
v
F
A
Phdn I chung t&i xet 3 ba.i toin Cauchy cho phtiong trmb Poisson trong
dl3 troll don vi Dc R2.trong nU'am~t phAng tren p+ c R2. va. trong mta kh8ng
ih
glaD tIeD R\ ; v(Ji dfl ki~n Cauchy (u. c7v- d~o ham theo htidng phap tuye'n
ngoai tren bien cua mien) du'<;1ccho tren mQt ph~D bien cua mien.
CI}th~. chung t3i Ian lu'<;1tchuy~n dc ba.i toan kh3.o sit ve vi~c giai mQt
phtiong trmh rich phh Fredholm lo~i mQt :
Av=F
(1)
trong d6 A la. mQt loan tU'A:H
~ Hi. v(JiH. Hila bai kh8ng gian Hilbert. Trong

tung bai toan . H. Hi HI.cac Hong gian Hilbert khac nhau.
Cdc dOnggdp mdi cua Lu4n dn Ii :
1) Chung minh dti<;1cding Ala. toaDto' tuye'n tinh lien tvc tu H vao HI .
trong d6 H va. HIla. hai khong glaD Hilbert (thay d6i rhea tung bai loan ).
2) Chung minh du'<;1cding ve' pHi F cua phu'ong trmh (1) tbuQc Hi ; (}
day F du'<;1cxac dinh tif cae dfi' ki~n eho tru'ck,
3) 86i vai hai bai toan san cua phan I chung toi da:du'a ra du'qcdanh gia
d6i voi chuan ~IIH >HI
Trong phan II chung toi xet bai toan Cauchy cho phu'ong trmh Laplace
trong t~ng g6 gh8 cua R3nhu'san
D = {(x,y,z):-<X) < x,y,< <X),0 < z < ~(x,y)}
vdi <\IthuQc Wp CI(R2),
Bli loin la llnl ham di~u hoa u trong D. li8n tl}clrong b. vdi u. u. . uy.
u, cho tru'oc tren ph~n bien cua D dti<;kbien dieD b(}im~t Z =$(x.y) .m.i to<ln
-2 -
nay la m5 hlnh R3cua bai toan da:du'qc khao s.h (xem D.D.Ang. D.N.Thanh &
V.V.Thanh: H Regularized Solutions of a Cauchy problemfor the LAplaceequation
in an irregularstrip", :Tournalof integral equations and Applications, Vo1.5, N2.4,
(1993), p,p. 429_441),
B~ng phu'dng phap Green va ly thuye't the' vi, chung t51 da: du'a du'<1cbal
toaD Cauchy d teen v~ phu'dng trlnh tich phftn d~ng tich chip san dfty d6i v<'l an
ham v(x,y)
=u(x,y.O) (la di8u ki~n Dirichlet teen bien z=O).
1
(~v)(x.y) = F(x,y) ; V(x,y) E R2 (2)
k
trong d6 G(x,y) = (Xl + yl ~k2)
"
Ie 13.h~ng s6 du'dng du l<'n. tho a
Ie> 4>(x,y) ,

V(x.y) E R2
Sd dl}ng phu'dng phap chlnh h6a Tikhonov (xem A.N.Tikhonov and
V.Y.Arsenin : Solutions of ill-posed problems. Winston. Willey, New York,
(1977», chung toi xiy dlfng mQt phu'dng trlnh bie'n pMn (phu'dng trlnh chInh
h6a) (00) :
~ Ve =Fe
(3)
Trong d6 bai toaD too nghi~m v=v" da phu'dng trlnh (3) la bai "toaD
chlnh, nghia la
i) T8n t~i duy nha't v"thoa (3)
ii) v" phI}thuQc lien tl}c vao Fe
E>6ngg6p quan tn;mg khac trong Lu~n an Ia chUng t5i dii dauh gia du'<1c
5ai 56 giU'anghi~m chlnh h6a v" neu teen so voi nghi~m chinh xac v cua phu'dng
trlnh (1)
-3 -
Cl}the la ne'u sai s6 giiia dii ki~n dod."eF£va dU'ki~n ehinh xac F la &
,nghlala
~F,-FII< Ii
(4)
thl eh11ngt8i eh1fng t6 du'<;1ela sai s6 giiia nghi~m ehlnh h6a v£va nghi~m chinh
xacv (Vdi~iathie'ttrdnthichh<;JP)C6b~C,fS hay
[l{~)r;(0<&<1)
nghlala IIv£-vll < c,fS (5)
II v vll < c[tr{~)r
hay
(6)
trong d6 h!ing s6 du'dng C kh6ng phl} thuQc S Ta chuiln 11.lIl1y trong cae kh6ng
gian tu'dng 1fng .
Hdn the' niia, ehl1ng t8i thi~t l~p dU<;1f;thu~t roan Giai tlch s8. Cl} the; nhu'
sau:

a) £>6ivdi cac bar roan khao sat trong phh I, chung toichd'ng Minh
du<;1erhg v.chinh la diem b1t dQngduy nh:ltcua mQtroan ttl'co thieh h<;Jp.Do d6
de dang dy dvng mQtthu~t roan l?p M tinh xa'p xl v£. O9i v£(rn)la budc l~p thd'
m .Chung toi da dua ra dU<;1cdaRb gia sai s6
Iv,(">-vl< C,k'" +C~ (7)
C£ 13.h!ing s8. phl} thuqc s. kh8ng phl} thuQc m . k E (0.1) 13.h~ s8 co. Hdn niia
ne'u chQn budc l~p t6i thieu m=m. tIll chung t8i thu du'<;JcdaRb gia sai so'
~v}",) - vii < (1 + C)J;:
(8)
b) £>6ivdi bai roan trong phh II, chung toi du'a ra du'<;1ccong th1fctu'Clnp,
minh tinh v. theo dU'ki~n do d~c F£ thong qua bie'n d6i Fourier (hai chi~u) thu~II
va ngu<;1C.Vdi gia thi6t v du trdn (v E Hl(R2» chung t8i thu du'<;JcdaRb gia sai s6
-4-
1\v vll < C[~;)r
I"~
trong d6 h~ng s6 C chi ph'} thuQc vao Ilv~lh'11)
Lie ke"lqua cbillh CIIaLlI~ll all (hi<,lccong b6 trong [1] .[2] va se
c!til/ccong b6"lrong [:\J.[4J.[SJ"
tJ
-5 -
PJIANM6r
cAc sAI loAN CAUCHY
CHO PHUONG TRINH POISSON
I. BM roAN CAUCHY CHO PHtJdNG TRINH POISSON TRONG HINH
TRON BdN VI :
1.Bdi loan
G~i
D =I(X,Y):X2+l < I}
15 = I(x,y): X2 + y2 ~ I}
Tlm ham u = u(x,y) th<Saphu'dng tr1nh Poisson'

!!.U= f trong D (9)
u E C2(D)nC1(15)
va di\!u kic$nCauchy du'<1ccho tren mQt phh bien ct1a D nhu' sau :
U(CO50,5inO) = uo(O)
~:(COSO,5inO)=Ul(0) O«}<a (10)
vdi f cho tru'dctrong D; Uo . III cho tru'dc tren r = {e'o:0 < 0 < a} .a
t3u
cho tru'dc Q< a < 21f kj hic$u
- la d~o ham theo hu'dng pMp tuyen
. t3n
n
=(c~O ,sinO) tren t3D hu'dngra ngoai d5i vdi D
2. Thitt l/jp phr/dnll lrinb deb phlin
iJu. . t3u
Ch9n v(O)
=- (cosO ,5mO) ,Q ~ 0 ~ 21f 13.an ham, () d~y - la
t3n t3n
d~o ham theo hu'dng pMp tuyen ngoai tren vong troD ddn vi 13D.
- 6 -
B~ng phu'dng phap Green chung toi du'a bai toaD (9), (10) v~
phu'dngtrlnh tich phan Fredholm lo~i mQtd6i vdi /inham vnhu'sau :
2r
I
I - B
I
J v(/)ln2sinTdl::: F(O)
a
vdi F(O) :::1T[U(O)- uo(O)] -
Ilit (I) In21sin I ~ 0 Idl
+ ~ Iff(~, 1}){21n[(cosO - ~)2 + (sinB - 1})2] -In(~ 2 + 1}2))d~d1}

D
( 11)
(12)
3.Khiio sat phlidnf!. trinh tfch vhlin-,-
Rtf di 1.1 Ne'u lIo,U1E L2(0,a) va f E L2(D) tIll FE L2(0,a) vdi
F xac dinh CI(12).
I
1- 0
I
';jOE(O,a). hamt~ln2sin2 thuQcL2(0,21T).
. 2r
r
I-O
j
£)~t (Av)(O)::: J v(/)ln2sinTdt
a
thl ta c6:
Bil tfi 1.2:
(13)
Ml!nh tfi 1.1: Toan tU'A:L2(a ,21<)~ L2(0, a) la loan ttl' tuye'n tinh lien t\lC .
3. Chinh hOa nghiDm:
Xet phu'dng trlnh
Av ::: F (14)
vdi FE L2(O,a) cho tru'de va toan tU'A xac dinh CI (13). Nghi<$m v tlm
trong kh6ng gian L2(a ,21T)
Chung ta chlnh h6a nglll~m cua phu'dng trlnh (14) theo phu'ong
phap Tikhonov.
- 7 -
Vdi P > 0 va FE L2(O.a)
vfl E [}(a,2TC) saoeho

eho tru'de x~t bai loan : T1m
P(\'p,ffJ)+<AVp,Af!J> = <F.AffJ> ,Vf!JEI!(a,2f'l)
(15)
trong do ( . , .) va <.,. > Ih lu'~t la tieh vo hu'dng trong L2(a.2TC) va
L2(O.a). Chung ta ky hi~u cae ehuin tu'dngU'ngla 11.11 va 11.11 .Ta co ke't
H HI
qua:
Dillh Iv 1.1: Vdi m6i P > 0 va FE L2(O.a) phu'dng trlnh (15) co cluy
nha't mQt nghi~m vp E L2(a.27r) , hdn m1a vp phV thuQc lien t1}c vao
FE L2(O.a).
Ghl sU'Vo13.ngill~m chinh xac ella phu'dng trlnh
Avo =Fo
thoa di~u ki~n : T6.ri t~i v E L2 (O.a) sao cho
(vo.ffJ)=<v.AffJ> , VffJEL2(a.27r)
(16)
(17)
Killd6 ta co
Dinh It 1.2:
GiasltF.FoEe(O,a) thoallF-Foll <Ii vavo thoa
HI
(16) ,(17). G<;>iv, lil nghi~m da phu'dng trlnh bie'n phan (15) rl'ng vdi
p =£ thl ta co daub gia
livE - VO~H < Mii
M=C+I~U~.r
(18)
trong do (19)
5. Phlidnf!. IJhti~ s(J:'
XtSt phu'dng trlnh bie'n phan (Ii > 0) :
£(v.,ffJ)+<Av AffJ> = <F.AffJ> ,VffJEL2(a.27r)
(20)

hay tu'dng du'dng
- 8 -
1>1' +;\*;\1' =;\*r
F. B
(2\ )
ludo
I' =\' -n.
(
1>1' +;\*;\1' -;\*r
)
F. F. f' F. F.
v(fjfJ > 0 sc ch9n sau,
(22)
V~y vI>=T vI>v8i T: L2 (a.,27t) ~ L2 (a.,27t)
du'(jcxac djnh nhu'sau :
Tv= v-p(ABv-A *F)
(23)
() day
, *
AB =;E.ld+A A ,
vii ld - to<ln tarddn vi trong L2 (a.,27t)
Ta co ke'l qua sau :
(24)
Dillh Ii 1.3:
I>
V8i P=(E+ IIA If Y thl T Iiiphep co trong L2 (a.,27t)
He Qua1.1: 'liE:>0 cho tmac, phu'dngtrlnh (20) ho~c (21) co nghi~mduy
nha'l VBE L2 (a.,27t)
Ta linh VI>bAng phu'dng phap xa'p xi lien tie'p
(m) -

T
(m-I) -
12
VB - VB m - , '0"
v~O) E L2 (a 27t) tily y
Taco
(25)
,,~m)= (/- PEl v~m-l) - pA *(A 'J~m-I)- F)
v8i fJ nhu'trong Djnh Iy 1.3
(26)
Mellh d~ 1.2:
Gia sarv~ thoa (16), (17). Khi do sai s6 giii'a v~m) va vo 13
Il
vlllll - v
II
< (' kill + M r;
I> 0 /}ra 21t) I> v'<'-
(27)
- 9 -
IITv.(0)-". (0)11,
d da C = L(a,2") ,
(
28
)
Y . I-k '
k - h~ s8 co eua anh x~ co T ; (0 < k < I) va M de djnhb"l (19)
Ml!nh dO1.3:
, ~f)
ChQns6tVnhl~nm. > Ink
Bat v = v (M,) khid6

.
Il
v - VoI, < (1+M)JE
, ilL(a,l )
(29)
(30)
Cilu tb.ie!!.;.m.la s8 bd&:l~p t8i thi~u di ta e6 danh gia teen.
MQtph~n ket qua eua mvc nay dii ddcjcc6ng b6 trong [I] va[2].
II. BAI ToAN CAUCHY CHO PHVcJNGTRINH POISSON TRONG NUA
MAT PHANGTRtN:
1. Biz; loan:
GQi
p+ = {(x,y): -oo<x<oo,y>O}
]5+= {(x,y): -oo<x<oo>y~O}
Tlm U E Cl(P+)nC2(]5+) .Uy E C(P+)
Au = f trong p+
u(x,O) = uo(x)
Uy(x,O) = UI(x)
thoa
(31)
"Ix E 1= (-1,1) (32)
u ehinh qui d v8 eung. nghla Iii.t6n t~i h~ng s8 dddng B sao el1o
lim sup u(x,y) = U'"
R-++a> x'+,,'-R'
y>o
B
IVu(x,y)\
~ Xl + l
'V(x,y) E P+ vhl + l duMn
(34)

- 10 -
~ diiy Vu - gradient cua u.
f chotn10ctrongr .Uo,u. tho tru'<'1ctrong (-1.1) ; Uy- d~o ham rieng cua
u theo y .
2. Thitt llip p1uh1nlltrinh tlch phlin
Ch9n v(x) =Uy(x,O), x E J=R\I= {x:~1 ~ I} lam in ham.
B~ng phu'dng ph3.p Green, chung tBi du'a bal to~n (31).(32). (33),
(34) v8 phu'dng trlnh tich phan Fredholm lo~i mQt sau dAy d6i v<'1ilin ham
v(x)
v<'1i
Jv(~)~- ~Id~= F(x)
J
F(x) = 1T(Uo(x) - u,.,)-
Ju1 (~) lntx- .;Id.;
-I
- ~ If 1(';",) I {(x- .;)2 + ,,2 ]d~d"
(36)
(35)
3. Khdo sat phJif1n6, trinh tic" phlin
i)
ii)
Hi)
Gi:lthie't:
UO,UI E L2(J)
.1=(-1.1) (37)
f ~ L~(P')
={rI[II'« ,.)f'«, .)<1<}. <w} (38)
vdi 1l(.;",)=(I+I.;I+I"r2l1; () >0 chotru'&
v EL~(J) = {v:[ p(~)v2(.;)d.; < cO}
voi P(~)=(1+1.;~2

(39)
- 11 -
Ai! 1.3:
Voi
e>0 chotniOc ,-1<x< 1 ham If'x:p+ ~p
~n[(X-C;)2+1]2]1
If/Jc;,1]) =
{
~
}
I+II
1+[(X-C;)2 + 1]2]
tbuQc I} (r) . Hdn nua t3n t'l-ihJing 86 du'dng Bt =Bl (0) saD eho :
Ilv/JL,(P')~B1 'l/xEI
BUdn 1.4:
Voi cae gii thi~t (37), (38) ta e6 FE L2(I), (Jdily F xac dinh theo
(36).
Mellh di 1.4:
VvEL2(J) ; p(x)=(1+lxI)2
B~t (Av)(x) = Jv(C;)1n!x-c;ldC;
(40)
J
Khi d6 :
A: L~ (J) ~ L2(I) 13.loan t:U'tuy~n tinh, lien t1}C.Hdn m1a
11.411~ 6
~':fnh}'f)a nRhMm:
Phu'dng trlnh c6 d'l-ng : Tlm 'liv E L~(J) tho a All = F
r FE L2(I) eho tru'dc va toan td' A de dinh theo (40) trong
Vdi Ei> 0 xet phu'dngtrlnh bi~n philn
tu'dng du'dng

hay Ia
vC1i
liV. +A'Av. =A'F
£v. +A'Av. -A'F = 0
v, =v, - P(EV, +A'Av. -A'F)
/3 > 0 se ch<;msau
(41)
(42)
- 12 -
V~y Vs =T jIB VOlloan tifT du<jcdinh nghTanhu sail:
2 2
T:Lp(J)~;Lp(J)
T,,=v-p(Asv-A"F)
i'1day
As
=E.ld + A" A
va Id - loan tu don vi trong L~( J)
Dillh Ii 1.4:
VoiP= E 2 thi T 1aphep co trong L~(J)
(H36)
(43)
(44)
He Qua1.2:
\::IE>0, \::IFE L2(J) cho tru'oc phuong trlnh (41) co nghi~m duy nhKt
2
vsELp(J)
Giii su rnng phuong trlnh
Avo =Fo
co nghi~m chinh xac Va san cho t6n t':li v E L2 (I) thoa
(vO,q»L~(J)= (It; Aq»Ll(/) \::Iq>EL~(J)

(45)
(46)
Ditlh IV1.5:
Giasu va thoa (45), (46) va !IF- Fo 1~1(/)< E khi do
lIvE- Vo II~(J) < M F
i'1 day Va -nghi~mciia phuong mnh bie'nphan (41), cfing ill di~m b1lt
[
2
]
1/2
. 1+ 111,111
~? . L(l)
dQngcua T, M = 2
5. PIll/dill!vM,} sri:
Ta tinh jIB bang phuong phap xa'p xi lien tie'p
,(111)-
7
' ,(111-1) -
I
"
Is - IE m- ,-,
(0)
1
2
(J)
, ,
Vs E 'p tHYY.
- 13 -
6
Chon

jJ
::: thl
. (6+36)2
r- 62 1
v;" L i. (E ~36)T 1';",.1)- (c +C36)2A'(Av~" ) - F)
Khi d6 ta co hai mc:nh M (I ') v~ 1.6) IIMnp,ht vai hai mt$nh d~ 1.2 va 1.3 d
Inl)CI. MQI phau k~'lqua Clla lIJlle !Jay se ch(yc caug b6 trou!?,131
.
(48)
Ill. BA.ITOA.NCAUCHY CHO PHUONGTR1NHPOISSON TRONG NUA
KHONGGIANTR.t:N:
Llld{.O!I.T1':'
f)~t
R;::: {(x,y,z): -00< x,y< OO,Z > a}
J{3::: {(x,y,z): -00< x,y<oo,z z a}
Q::: {(x,y): X2 + y2 < I}
T1m ham u E C2(R;1)nC(R;), UzE C(R/) thoa
t>.u:::f trong R;
(49)
vdi dil ki~n Cauchy du<;1crho tnf(1c tren dla Iron ddn vj et1a m~t phang z==O
u(x,y,a)::: uo(x,y)
11,('1",)',0):::U,(x,y) V(x,y) E Q (50)
trong d6 f cho Iniac trong R; ; uo,u. cho tru'<'Jctrong Q ; IIz d~o ham rieng
nla u theo z .
II chlnh qui d v() rUng. nghia la
r 1
l~d ,,+~+~)lI(x,y,z)l
j
:::a
L z>o

(ii) T3n t~i hhg s6du'dng C sao cho
I
vu(x,y,z)I:<o: C v<'JiX2 + l +zz dtl Wn
_,,2+y2+Z2
(i) (51)
(52)
2. Thf.J.11!1JfjP,1!l!.1idrlf!.trinh tfch phtin:
. 14 .
Ch<?n v(x,y) =uz(x,y,O);
(x,y) E Q = Rz \0= {(x,y): XZ+yZ 21}
lam an ham.
Bhng phtidng phap Green. chung t8i dtia bai toan (49). (50), (51).
(52) v~ mQtphu'dngtrlnh tich ph~n Fredholm lo~i mQtdo'i vdi :in ham v(x.y)
nhu'san:
II
v(~,17)d~d17
J. Z Z F(x,y) . (x,y) E 0 (53)
Q (x-~) +(Y 17)
II
UI(~ ,17)d~d17
F(x,y)=-21fUo(x,y)- J
a (X-~)Z+(y-17)Z
-HI f(~,17,t;)d~d1]dt; (54)
R?J(x-l;)z+(y-17)z+t;Z
vdi
3. Khiio sat pbrldnll. trinb dcb phlin:
Gia thi€t:
(i)
(ii)
2

Uo E L (0) (55)
Ut bj ch~n trong Q (56)
f EL: (R) 0 {fo Iff p( x,y ,z) f' (x,y,z )dxJydz < ro}
. vdi P(X,y,z)=(l+Jxz+i+zzY (57)
(Hi)
Bil di 1.5:
V(x,y) E 0 tich ph~n soy rQng
II
dl;d17
J(x,y) =
Q J(x-~)Z +(y- 17)Z
hQit~ va J(x,y)::;; 41f
Bil di 1.6:
- 15 -
Xet ham 'l/x.y:Q -) R+
(x,y) E 0 Ia tham s5
I
Ij/xr<I;, 17) = r: 2. z J 2 2' vII-ex-';) +(y-11) (x-';) +(Y-17)
fhl '(1',1E L2(Q). Hdn m1a
~
II
11
2 2- ~x +Y
VI'. ~ 21r In
x;, Lt(Q)
I
J2 2
- x +y
!1ff !li1.7;
Xet ham x.,,:RJ -)R+ ;(X,Y)EO lathams5

I
X.,/c;,17,C;) =
{
J 2 2 2
}
J 2 2 2
1+ (x-C;) +(Y-17) +C; (x-I;) +(y-17) +C;
thl X E I} (RJ) .Hdnm1a
Il
x ~2 ~ 1r
(
2+~+ ~lil2
)x" + ."IILt(R;) 2 2
IJ6 atLJl-,,-
B~t F1:O -)R+
F
(
x
)
=
ffJ
If (I;, 17,ola,;a"ac;
"y J 2 2 2
R; (x-';) +(Y-17) +C;
. 2
ncu fthoa (57) thl F, EL (0)
Mff.lllLdiLZ~
Gi;l sU' Uo thoa (55). u1thoa (56) vafthoa (57). Khi do. F xae dj.nh
bdi (54) thuQc L2(0)
Menh di 1.8;

VJi roM 'EL:(Q) =!, II w«, "),' «, ")d91" < 00)
trong do W(';,17)= 1+ J,;2 + 172 (';,11) E Q
- 16 -
If
,,(;;.rt)d;;drt
J)~[ (;1,,)(x,y) =
Q ~(x-;;)2 +(y_rJ)2
Khi do A:L~.(Q) ~ L2(0.) lit loan ttYtllyc'n tinh lien t\le.llon nITa
(
16
)
1/2
11;1lis 1t 3 In -;-
4. Chlnh hOa nehiem:
Phuong trinh (53) co d~ng
Av= F
Leongdo FE L2(0.) cho tru'oc va loan ttYA xac dinh theo (58).
(58)
(59)
Xet phuong trinh bic'npMn LeongL~(Q)
'" '"
EVE+ A AVE =A F
Khido ve =T vevd'itminttYTdulfcdinh nghIanhusau:
2 2
. T:L (Q)~L (Q)
Tv= v-I3(Aev-A "'F)
P> 0 se xac djnh sail
'"
Ae =E./d+A A
/d Ja loan ttYdon vi tTOngL;. (Q). Chung Laco kc't qua

(60)
(61)
Dillh 1i 1.6: VOi 13=
(
E
)
2 thlTJaphepcotrongL~.(Q)
2 16
E+ 31t In

e
Hequa 1.3:
V t:> 0 , V F E L\0.) eho [rudc, phu'Ung trinh (GO) e6 nghii;m tiny nhi;[
2
liB E L (Q)
Bay gio gia stYriing phuong trinh
Avo=Fo
eo nghii;m ehinh xae 1'0san cho t5n ~i i' E L2(0.) thoa
(v().<p)
= (17;;1<p) V<pe L~.(Q)
(63)
(6.n
- 17 .
ddiJy (. ,,) va <.,.>I~nlu'(,1tlatichvBhlf<3ngtrongL:(Q) vaL2(O).
Dinh IV1.7 ~
Gia st!' Votho a (63), (64) va If - Fnll<s .Kbl d6 t6n t,!-ih!l.ng s6
dlfdng M thoa
Il
v - Vo
II

' < M J6
, L,«I)
( 11111'11£,(0) )1\ "
v<Ji /0.,1=l ; ) va v. la diem b:it dQng cua anh x'!-co T dng v<1iG
(t{(c v, Iii nghi~m daphlfdng trlnh bi€n phan (60»
5. Ph"dn~s{l:
B€ til1hdi€m bit dQl1gv. ctia anh x'!-co T chung ta dung phu'ong
phap x;i'pXl lien ti€p
v(",)= T\,(",-I) , m = 1,2"
. .
v;o)E L~(Q) tuy Y
s
Chon
fJ
=-, thl
.
i 16
)
2
l£ +311'21n~
2
(m)_
1
S (",-I) s
A
'
(A
(m-I)
F
'

)
v.
- -
l
' 16
)
2 v. -
l
' 16
)
2 Y.'-
s+311'21n~ &+311'21n~
(65)
(66)
Khi d6 ta c6 2 m~nh d~ (1.9 va 1.10) tu'dngtv nhu'hai m~nh d~ 1.2 va 1.3 J
m1}c1.
MQt phan k€t qua cna ml,1cnay se du'(,1crang b6 trong [4].
(~J
- 18 -
, .
PaW HAl
BAI TOAN CAUCHY CHO PHVdNG TRINH
JI. ' '.,
LAPLACE TRaNG TANG G6 GH~ CUA R3
1. IJAI TOAN
B~t D={(x,y,z):-co<x,y<co,O<z<fS(x,y)}
15= {(x,y,z):-co< x,y< co,O:<:;z:<:;;(x,y)}
vdi 4>thuQc ldp C1(R2)
T1m ham UE C2(D)rC1( D) thoa
.61-1=0 trong D

ux(x,y,9'{x,y))= f(x,y) ,
uA x,y,qj(x,y)) = g(x,y)
uA x,y,qS(x,y)) = h(x,y)
u (x,y,qj(x,y)) = u1(x,y)
vdiJ. g . h . Ul cho tru'dc trong R2.
(67)
(68)
i'
Bay Ia bai toaD Cauchy cho phu'dng trlnh Laplace va nhu' chung ta
da: bitt 111.bai toan kh8ng chInh.
2. THANH LAP PH(j(JNG TRlNH TicH PHAN
Ch9n v(x,y) = u(x.,y,O)lam lin ham (-oo<x,y< 00)
X6t ham Green cua bai toaDDirichlet cho phu'dngtrlnh Laplace
trong nua khong gian tren
G(x,y,z;~,17,0 = r(x,y,z;~,t'/,0-r(x,y,z;~,t'/,-0
- 19 -
v~i f(x,y,z;.;, 1],0 = 1 ~' 1 la nghi~m cd ban
4Jr (x-,;y +(y_1])2+(z-02
crlaph1.fdngtrlnhLaplacetrongkhonggianR3; (69)
Gia thi8t :
i)
:(x,y)= :(x,y) = 0;
v~i r = ~X2+y2 dll l~n
(70)
ii)j{x,y) , g(x,y) , h(x,y) , /lb,y) d4n v~ 0 dll nhanh, nh1.f
- ~ khi ~X2 +y2 ~ ro (71)
\jX2 + l
ill) JI+x2 + y2 .v(x,y) EL2(R2)
(72)
iv) u chinh qui <"vo cung, nghla la t8n t~i h~ng s6 d1.fdngA thoa

A
\u(x,y,z)1 S;
JI+x2 + y2 +Z2
I
Vu(x,y,z)
\
s; 2 A 2 2
1+.1' +y +z
khi ~X2 + y2 +Z2 dd 1~n.
(73)
(74)
B~ng ph1.fdngphap Green, chung ta nhb M<1cph1.fdngtrlnh tich
ph~n sau v~i ftn ham u(x,y,z) va v(x,y)
1 "'
J J
'" zv(t;,1])d.;d1] -
( )
- M-ux,y,z
2Jr-a>-<o[(X ;>2+(y- 'f'J)2+Z2]
- J J O(x,y,z;.; ,I], ~(.;, 'f'J»)1. (.;, 'f'J)d.;d'f'J
-<XHD
- 20 -
+ J J GJ¥,y,z;~, 1],t}(~, 1]»)Ut(~, 17)d~d1]
(75)
ddily
0 0
f (~,tJ)= h(.;, tJ) - f (.;,1]) ,p(~, 1]) - g(.;,1])-;- ,p(.;,17)
0'; (/1]
va G1(x,y,z;~, 1],(J(~, 1]»)= Gt;(x,y,z;.;, 17,(J(~,1]»)
0

-G,(x"v,z;.;: 1],,p(.;,tJ»)- ,p(t;,tJ)
at;
0
-G~(x,y,z;1; ,1], ,p(.; ,1]»)- ,p(1;,tJ) (77)
01]
Cho z ~ ,,(1;,1]) trong (75) chung ta nMndu'c!c phu'dng trlnh tich
phan vdi ftn ham v(x,y)
(76)
~
JJ
,p(.;,1])v(,;,tJ)d.;dtJ -~u x
[
2 2 2
]
~ - I ( ,Y)
211:-<r>-CD(x - t;) + (y - tJ) +; (';,1]) 2
- J J G(x,y, ;(x,y);.;, 1];;(t;, 1]»)f (1;,17)d.;d1]
-aHX>
+
J JG1(x,y, ;(x,y);.;, 1];(J(I;, 1]))UI(1;,17)dl;d1]
i'
3. THANH LAP PHUONG TRiNH TicH PHAN CHAP
1
JJ
zv(~,17)dl;dtJ
Ham H(x,y,z)
=-
]
1/ di8u boa trong
2

[
2 2 2'"
11: -aHX> (x - 1;) + (y - tJ) + z
nd'a khl>ng gian tr~n z> O.
E>~t
A(x,y) =H(x,y,;(~ ,1]»
oH
p(x,y) =-(x,y, ,p(l;, 1]»
on
(80)
(81)
- 21 -