giải phương trình laplace trong miền ngoài của quả cầu

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GIAI PHUdNG TRINH LAPLACE
~,

,.,

.,

~

TRONG MIEN NGOAI CUA QUA CAU
.,

'

I. Ma DAlJ:
Xel b,li l()(1ngia lri bien Dirichlel cua phlWng ldnh Laplace d6i vdi mot
mi6n D (lIdl1lien va bi ch~n):
LlU(X)= 0, \iXEW\i5
HeX)

trong lit') r 1:1ham s6lien

=rex),

LlXEaD

t~lc tren aD.

-DE chC(ng minh gl)n ma khong mat llnh l6ng quc1t,trong Chl(elng nay
chung loj xel mi6n D la qua diu deJn vi B={XER" : x < I} vdi bien la a13={
I

I

x E R": x < I} trong khong gian RII vdi n~ 2.
I

I

-Bai loan gia lr! bien Dirichlet mi6n ngoai se c6 nhi€u nghi~m.Trong
cac Sc1chgiao khoa,ngl(oi ta thl(Ong them tlnh chat CI}th~ cua u 0 vung vo q(c
lhl mdi xac lIinh dl(l}C bi~u thac cua u.Trong chuang nay ,chung toi l1l(a fa
ml)t dang nghiQI11u1ng qm1t dll cIll(a 1'0tinh ch~fl cua 1I0 vo c~(c.
-N(ji dung cua dll(ling nay dlCl;5C
tdnh bay d~t'a theo bai ban IS].

II. lYIQTs6 DJNH NGHIA vA DJNH LY .
1. Dilll, if' :
Xel mi2n 0

= 13 (0,1')

\ {O}

= {x

E

nn : 0 < x < r}, vdi r> D.

Khi lIl) lc1nlc.lih~ng so c c6 cae tlnh chat sall :

. C E (0, I)
. Ml)i h2\1111i6uhoa
c

dlCOngu xac l1inh tren Q d€u th6a

\ix, Y E 13 (o,~),

Ixi = Iy\=> clI(y) < u (x)

Clul thich:
Hang sf) c khong ph\! thuoc ham di6u boa u.

Clllfllg mill"

-

Gl)i K IA m~t ceiu ban ldnh ~ (tac K

=~

S) lhl K la t~p col11pacttrong

O. Theo l1inh 1)/Harnack thl tl1n t~\i h~ng s6 c E (0, 1) co tinh chat


J;L(1 It 'jrt~t-(&to dt;c

37

200/

C U(Y)

Jfiu;}IbJt- ,9'h{Wth-

%

< U(X)

VtJi nH}ih~lIndi~ll hoa dl((jng II xac djnh tfen Q va v6i x, Y E K.
Cui t lllY Y thuoc (0, 1) va u 13ham di~u hoa dlf0ng tfen n.

-

H~l1n s(1 x -1- u (tx) cung 13 ham di~u hoa duong tren n, do d6 eung
tho a bat dang lh(ic 0 tren, t(ic la
CU(ly) < u(tx)

i

M6i x E

S se tl(ong (ing duy nhat mot ph~n to' tx E ~. S, do d6

eu (y) < u (x)

- Ta thay

r
'\Ix, y E - S
2

,

tr
'\Ix, Y E - S
2

,

~ c6 lh~ c6 gia lrj bat k5' trong khoang (o,~)

ne'u t dtiQC

chqn thich ht.ip trong (0, I). Do d6

'\Ix, Y E B (o,~),

I

x

I

=

I

y I=>eu (y) < u (x).
0

2) Villh If':

.

Gi~i sli II : Rli -1- Ria ham di~u hoa tren B\{O}={XERII : 0<

.

Ham sf) A[ullren

Vdi x tuy

Y

I

x < I}.
I

B\{O} ducjc d1nh nghla nhl( sall:

thuQc B\{O}, A[uJ(x) la gia tr1 trung blnh eua ham u tren

m~t diu b,1n Idnh lxi, tuc la
Alul(x):=

twile

111-l fu(y)dS(y)
mix!
lyl=lxl

AluJ(x) := ~ fu~xl~}lS(~)
1~I=l

trung lit) w la di~n rich cua m~t du don vi.

.

Khi l16 AluJ co bi~ll thue nhu sau :

--.

Tncc'lnghejp n = 2:

.


l

.Yiui /1 'Prl~t' ~o

~~c

38

200/

Alu!(x)=b(-Inlxl)+c,

./1j;?J/Zn

,9'~tV/t~

'r:i

VXE B\ {O}

trung d6 b, c IA h~ng so.
-

TnCongh(Jpn > 2 :
Alul(x)=b

IxI2-11+C,VXEB\{O}

trong do 11, IAhhng so.
c

CIll£llg lIlillh

Xet ham so f : (0,1) ~ R dlCxac
Vr E (0, I),

ref)

=

fll(r~)oS(~)

s

(1)

Sau day ta se ttm bi~u th((c clla f( x I), r6i suy ra bi~u th((c CI1aA[u](x).
I

-

Ham s<1-:,~ u(r~) lien t~ICd~u khi r~ thuQc t?P compact trang B\{O},
ur

00 d6 d~1O
ham cua f IA
1"(r)

= f~u(r~)oS(O
or
s

= f~. ('Vu)(r~)oS(~)
S

D~l bien moi ~ = r~ thl
1"'(1')= rl-l1.

-

dl(\jC

f;.('VlI)(~)dS(~)
rS

Cui rl), 1'1 tily Y thuQc

(0, 1)


(/z

.~o

J(t(t/{

L/;/,

39

>(I/It- l tJao O~9C 200/

Gqin = {x

E Rn : ro

f U.VlI

LiS:::: fllll
0

= roS u

trong d6 .an

~

-

'U=

Jt;~'JI&l/ 3h(Mt~ CY;i

< x < rd
I

I

Ap dl,lng cong th(I'CGreen,

80

.

ta dt((}c

dV

rl S
Bell

~E foS

ro

J

1

neB

1 rl

~E rlS

( U lAphap vecld drin vi hudng ngoai cua an ).
./'>U= 0 (do u la ham di~u boa)
Suy ra

f =-~. (V'lI)(~)dS(~)
~)
r()S

+

f 1. (\7u)(~)dS(~)
~

=0

1'1
S

f

Do d()

,

~,s

I~
0

.(Vu)(~)dS(~)::::

f

~S

l~
I

. (Vu)(~)dS(~)

Tl'( dJng lhere li~n tren, ta suy fa
'Vr E (0, I),

f~.
r
rS

(Vu)(~)ds(~)

= h~ng

sO' (h~ng sO'nay c1u~:!c Qi la kl)
g

- 1'1:(bi~u th(fc clh 1"(r) (j tren, ta SHYra

fer)

= k1 rl-n,

'Vr E (0,1)

Do d6
k [In r + k2

neu n:::: 2

1 k Ir2-11+ k)

1'(r) :::

neu n> 2

J

Ma Alulex) = ~f(lxl), SHYra Alul (x) c6 bi~u thac nhtI' phat bi~u (rang
0)
c1jnh ly 11.2 .


(p
,J..u(h,

'/:'(0
;/(/;/t

c.iJ/'
l-5ao Ol9C

.f

40

200/

-

~

'./(p';jIMb

('::5,-

o;/~

.!3hCVlth- f/ tt

0

3) Djllh Ii :
Gic1slt ham so th~(eu co cae tinh eha't salt:
i) Lila ham oi~u boa tren B\{O}={XE RI1:0<1x 1<1}.

ii) Lila ham lien t~le tren :B\{O}={XE
iii) u(x)=O,

RI1:o<1xl S';1}.

'\IxE8B={XERI1:lxl=1}.

iv) u(x)2:0, '\Ix E B(O,r)={xE RI1:0<1x I (0, I ) ).

Khi do u co bi~u thue nillt salt
-Tn((1ng h<;1p
n=2:

u(x)=k(-ln I xl),

'\Ix E B\ {O}

trong do k la h~ng so kh6ng am.
-Tnf0ng hejpn>2:
HeX)

= k(

1

X 12-111),
-

'\Ix E B\ {O}

lrung do k la h~ng so kh6ng am.

Chl'tllg millh

G()i Alul(x) la gia trj trung binh eua u tfen m~t du ban kinh x I.
1

Ta se eh(tng minh u = A[u] tren B \ to}.
Xel mi~n B(O, f), rhea dint ly 11.1 (ehudng 4) ton tc~ih~ng so eE(O, I)
saG cho vt'ii nH,>i
ham oi~u boa kh6ng am u tren B (0, f) o~u thoa

. cu(y)

< U (x)

'\Ix, y ~ B (o,~)

va

D~l W = u- e.Alu]

Chltllg ll1inh dW1e chiu thanh cae bttde nlll! salt.

Ihioc 1: (Chung minh w kh6ng am lfen B \ {O})
. Coi x E B ( O,~)

'\Iy E B (o,~),

Iy I = I x I => eu (y) < u (x)

I

x

I

=

I

y

1


,

I. . (/)

,'fWill

f/tZl?

..,y'
(.;YfO ekeD

200/

41

LA~~:yZlt

.9'h(l//bti'f/;i

SHY I'd cAlul(x):::; u (x)
lI(X) - cAlul(x) ~ 0

w(x) ~ 0
Do ell)w khC>ug lren B(D,~)
am

.

G(.>iL

== inf

{w(x): x E B\{O}}

l'{)n lai day {ad lwng B \{O} saD cho w (ad -7L khi k -7
Day [a"J BallI trung t~p compact

00

B nen t6n t~1iday con {bd hOi l~1v~ b

E D

Tn('1ug IH.fpbEB\{O}:
Ilam di~u boa w d'.ll clfc li6u khi x == b, lheo nguyen Iy c~tcti6u
lhl w la ham hang lren B\{O}, w c6 gia ld khong am trang
13(o,~)

nen suy ra L ~ O.

Tn('jng IH.fp
bEGB : w(b)=D nen L = D.
Tn(Ojugh(}pb == : w khC>ng lren B(o,f)
0
am

nen L ~ O.

Ta 11IC>u L ~ 0, do d6 w khong am lren B\{O}.
d)
V~y

la Cl) lI(X)

Ihide
-

-

(1)

cAlul(x) ~ 0, '\Ix E B \{O}.

2: (ChCrng minh HeX) - Alul(x)

~ 0, '\Ix E B \ {O})

Cluj \ lll: If! clay s6 dlfC}C
c1jnh nghla bai
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.

> 0, do d6 day {tm}tang, d6ng thai clay nay
hi ch~n biji I Hen hoi tl,lv~ gidi IH,ln ta gqi la 1.
ma
Ta U)

tlll

- tlll-J == C (1 - till-I)

Cho 111
.--)-00, ll( (5.2) SHYra
I ==c + t (l - c)


;;

"'/)
fUll

,~jtf(/(

tIll

',,/'
,)1oc

42

:..!{,i(J/

---)0

(3)

I khi m ---)0 .
00

Xel day ham {wm} yoi Will= U -tm

-

. /(y; ~yg1t . ?-ZUltlt 'Yci

=1

l

v ~y

'(Jew

Ta lh,l'y Wm - cA\wm]

=u -tm
= II -

A[ul

'

Alul-- cA [u -lmAlu]!

tmAIu I - cA Ill! + rImAIIII

=u-lc+tm(l-c)IAlunl
=U-tlllt-1A\Llml

= Wm+l

n'rcla

WII1+1 =wn-cAlwnl

- Thcl) kel qLl21 d bt(dc I thl ham di~Ll hoa Wo khong am tren B\{ ()}.
(I)
Ap dl,lI1gkc't LlU21I) Jai vdi ham W()thl Ol((,iC
(
tinh chflt khong am clla ham WI
tren13\{O:,
Bang Lluy n"Lp, ta suy ra ham Wm kh6ng am tren B\{O} vdi mQi so'
nguycn dll\!ng

111 ,

tLtCla

W11l(X) u(x) - tmAlul(x) ~ 0, \imE N ,\ixE B\{O}.
=
Clio

Il) -} 00 thl tm ---)0 1 (uo (3)),

u(x)-AllIj(X)~O,

sLoe

(4)

\ix E 13\{0}

IhiO'c 3 : (Chang minh u

Giil

suy ra

= AluJ)

It'\n ti;li Xu E B\ {O} san cho u(xo)

> Alu](xo)

Do u lien t~IC[(.IiXonen t6n li;limOt Ian c~n V clla Xosao cho
HeX)> A[u](xo) , \iXEV,

M~l kllac, theo ket qua (4) d bt(dc 2 ta co
lI(X)~ Alul(xo) , \ixEoB(O,l xol)
SHY ra gia trj trung blnh cua u tren m~t dtu 8B(0,l xol) iOn h(in A[lIl(xo),
llrc la

AIul(xo) > AI ul(xo)
Dietl n~IYvo 19.
Do lit) khong c6 Xol1aO thuQc B\{O} ma co trnh chflt U(Xo)> Alul(xo), ket
h(}p ydi tlnh cha't (4) ta dt«}c.
HeX)

= Alul(x),

\ix E B\{O}


I L'

. Lw(

, 1"

It

(//

Fltlto

l:;Y~

O{tO (7i9c

43

20{)/

Jf:f~VJlb;to L9'hCl/lth- %;

Ihifie 4 : (ke'llu~n)
Alul c() bi~u lhlrc nhl( lrong dinh 19 1I.2(d1lWng 4), nen u Lung y~y.
Tn((Jng

-

Ta Ct) u(x)

hl}P Il
co:

co:

2:

be-In

x I) + c, \lXE B\{O}

I

Khi Ix I-~ I lhlu (x) -) 0 nen suy ra C

Khi x

E 13 (0,

V~y u(x)
-

=o.

r) lhlu (x) ~ 0, suy fa b ~ 0

= b (-In

I

x I) ydi b ~ 0

Tnf(ing hl.ip n > 2:

Tau')u(x)co:blxI2-n+c,

\Ix E 8\{0}

Khi I x 1-) I lhl u (x) -) 0 nen SHYfa c
Llu lk) u(x)

=b (

I

co: -

b,

X 12-11-I)

Khi x E 13 (0, r) lhlu

(x) ~ 0, SHY ra b ~ O.
[J

4/ Di"" /y :
GiJ sLY
lulll I la Llaycae ham di~u boa lfen l~p m6 Dc U'\day nay hQi
It.!d~u vC:h~lmso u lfen m6i l~p B (a, r) c D.
Khi lit')u la ham aieu boa lren D.

Clll?llg milllt
Cui a IllY y IhuQc D

T6n 1<.li cau B (a, f) c Q
qua
'I'a s0 chl"rngminh u la ham ui~u hoa lren 8(a, f).
Do phep Ijnh Lie'nkh6ng lam lbay d6i Hnb di~u boa cua ham so nen ta
c() Ih~ gd sera= 0 ma kh6ng mill tinh l6ng quat.
Thc!) c()ng lh(rc Poisson (dinh 19 II.l-chlWng

Um (x ) co:

Ta Ihfiy

fH(~, x)
1~I=r

ul1l

(~)dS(~),

1) ta c6

'\lXE 13(0, f), \1m E N


,:
.Jtt(ilt

'-,,/'
f/tl/t co Clt~)C
C:7ao

Ulll (x)

-

'

'

44

200/

fH(~, x)u(~)dS(~)

Coi E > () ILlY

m>N=>

CP;i

fH(~, X)IUm(~) - u(~)ldS(~)

S;;

1~I=r

Jt;;~vyJ1b 3ha~th-

1~I=r

Y, do UIl1 t~1c1~uv~ U nen 16n t~li s6 nguyen N thoa
hOi

(IUIll(~)-u(~)I
Luc utI

LIlli(x) -

fH(~,X)U(~)dS(~)

S;; E

1~I=r

Do c1()LIlli(x) ~

fH(~,X)dS(~);;:o

E

1~I=r

fH(~,X)U(~)dS(~) ,khi m ~

ct:)

1~I=r

S UY ra

u(x)

;;:0

fH(~,x)u(~XJS(~),

\fx E B (0, r)

1~I=r

Do lit) U ui~lI boa tren B (0, r).
Suy ra u ui~lI boa tren B (a, r).
V~y LIdieu boa Iren Q.
0

5/ Hill" ly :
Gia Sl(LUmla chu6i cae ham di~u boa tren t~p m0 Qc R'\ehu6i nay hQi"
t~ll1~1I ~ ham s6 u tren m6i t~p B(a, r) c Q.
v
Khi de) II 1ftham c1i~uboa tren Q.
CIll?llg

11lillh

Binh Iy nay la h~ qua eua dint Iy 11-60 tren.
,

III-PHEP

~'

':"

BIEN DOl K.

1) Binh nghia:

.Xet l~p Ec R'\{O}.
V()i x la mot phftn tii'toy

y clla

E,phc1n tll x* du'<;je
dinh nghla nhl!' sau


__'fa'illi

f

t

45

{[~t «(0.0 (;7{~c 20()/

. -

cv

<-/'jtt;:jlbH ~h{1dt4

CY/.'
j/ ,t:

x*=~
Ixl2

T~p E* lh(.Jc oinh nghTa la E*={x* : XEE} .
2) Tfnh chftt ciia x*.
1
a) '\;Ix:;eO, Ix *1=

N

b) '\;Ix:;eO, x**=x
c)
d)

'\;lEe R"\{O} , E**=E
NeuxEE*thIx*EE.

e) Neu E la t~p h(.jp {XE RII: (k

I

x I

I >J I.

Cilling lIlinh
1

x
I

H) I' 'I = IH'

=

N
x
>

b) x**:=(x*)*=

x* - T = x
_Ixr

*2
x-'
I

1

Ixl2

c) Suy 1'a tu tint ChEllb).

d) Cui x toy

Y

thuQcE*.T6n t~i YEEthoa x=y* .

Suy ra x*=y** ,ma LheDtinh chat (b) thl y**=y,nen

x*=y.

VGy x* LhuQcE.

e) Suy fa lu ojnh nghTaclia x* va tinh chat (a).
0
3) Hill II IIghia.
Ghl Sl( L~pE e R'\{O} va ham so u : E-7 R .
Ham s6 KI ul au\ic djnh nghTa nhL(salt
Kluj:E*-7R
\;Ix E E*,

K[u](x):= Ix12-11 *)
u(x


(j:'
,cL,{ui/l

'/,"/)
;/(1/I"

Ciao

~~/'
O~~c

,//~ JlJtttJlbll-

46

:::!()O/

6,CY/,~
,!3h{b714 f/ c'i

4) /Jill" /y.
Gi~i su' Ii c IC\{ O} va ham so LI:E~ R .

Khi lh\bien d6i K cua ham so Kluj chinh la u ,tL1'ca KIKluj]=u .
l

Clll~ng minh
Do E**=E nen mien Xclcdint clla KIK[ull ehinh la E.
'IIx E E,

KIKIu II(x):= Ixl2-n Klu](x *)
=x1

2-11
1

2-n

"'

1

X'"

1

", ",

ux"""=ux
(
)

( )

V~yKIKlull=LI.
I]

5)/)i"" /y.
GiNc'u w=Klgl thl g=Klw]
CIll?llg millh
Gia si'i'w=KI gJ thl w :E*~R

.

Khi Lh)Kiwi c6 mi~n xac dinh la E**=E.
Vdi x lllY y lhuQc E,la c6
2-11

Klwllx)=

I

x

1

wx*=x
( )

2-11

2-11
I

1

I

x*

1

gx**=gx
( )

()

V~y Klwl=g.
[J

6) ViII" /y :
Gi~lSlt'u dieu hoa tren qua cau oejn V!B c R'\lien t~lC

tren B.
Khi (It) KIu I dl(JC xac dinh tren RU\B b(~ic6ng thue

~
KllIl(x)=

~

f

Ix12_1

neB

x ERn \ B

"' IQ=I!x - ~I" u(~)dS(~)
u(x)

.

neB

trung (M (I) I;) di~n tich clla m~t diu don vi .

Ixl

=1

( ,(

X'

...Lad/(

("L?

/(blt

-..i;/'

0uo

ilt9c

,//'

47

200/

-

J':ft':f/&t-

oT

cy/.:

J~(V1t;;

}/ /i

CJlll'Ilg millh
ThL:o cc)ng Lht'fc Poisson

La c6

I

u(x)=

1- x~-u(l;:)dS(l;:),\/

~ J

I

())

x E B.

Il;:- xll1

I~I=I

Mi€n xac dinh ciia KIHIla U!'\B (do Hnh chc1tIII.2-chl1dng 4).
-Khilxl=ILhl
KluJ(x)=1 XI2-IIU(X*)

=u(x)
-Vdi X IllY

LhuQc RI1\B , Laco

Y

1-11

x
11 -

KllIl(x)=

=

-

u(x*)

1

1-

1-11

Il;:l

X

Il;:

x 'If

-

1---

f

(0

1

u(l;:)dS(l;:)

=1

=~

*2

Ixl-

J

())

I

1

Ix I 2-11_lxI2

I~I=I

I

111

u(~)dS(~)

~ - --~-

Ixl2
l~e_= 1

=~

J

(0

Ixl2-n

Ixl2

11;1=I

=~

u(~)dS(l;:)

Ixll"l;\xl-I;1
Ix12-1 m u(l;:)dS(~)

J

0)

.

In

I~I=I ~Ixl-.-~
Ixl
1

2

ma khi

I

~I

=I

x
~Ixl- - = Ix - ~I

Ixl

Ihl

X

(do 1l;:lxl-N

?
I

= Ix -l;:I-)

nen
?

I
KllIl(x)=

-

J

x
II--I
u(l;:)dS(l;:)

(O1l;:1=IIX-l;:ln
0

( L'

.1Wilt

'I'

/?

j/~ht'

bao

~);/'

48

(j~(JC .2001

JJ:f;':fIblt ,-3Y;;a7t~
%

7) J)i"h Ii .
Gia sl'rt~p m0 E c IC\{O} va ham so u : E-t R .
Khi Ll6,ham so 1Idieu hoa (ren E ne'u va chl ne'u K[uj dieu hoa tren E*

Cillfllg millh

-BliO'e 1:
Giii sl'r1Ila ham so dieu hoa tren n.11.Ta

chCrngmint Klu] la ham so
dieu boa tren R'\{ O) nhLrsau.
Coi x lLIY thuQCqua du don vi B.
y
Ta cti u(x)

~

~

J I-lxi' u(i;)dS(~) JH(~.x)u(C)dS(~)
~

ffile:l=llx-ct

.

1e:1=1

Theo Llinh Iy 111.6(chLWng 4) ,vdi I xl >1 ta c6

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
Do LIt)Klujla

(theo chd thieh 0 Il-2-d1LWng I)

h~lJl1dieu hoa lren n!\B

.

Theo dint nghla,ta c6

'IXE R"IB. K[u](x):~ Ixl2-n{lx~2 J .
Suy ra Klulla ham giai rich tren W\{O} (do rich va hlJp cae
h~\111 rich la ham giai rich).
giiii
Do d6 ham ~(Klu}) la ham giili tieh tren UII\{ },dong thai ham
0
nay lrit;l tieulren R'\B,suy ra D.(Klu}) lri~t lieu lren eelR'\{O}, tue
ham Klu} aiel! boa lren R'\{O}.
-Ihide 2 :
XGt u 1[1
ham dieu hoa tren E c W\{ O}.
Cui b tLIY
YthuQc E*.
Ta c6 a=b* thuQc E thoa b=a * .
Theo djnh Iy IIl.6-clutong 1,t6n t~\imQt Ian c~n Va clla a saD cho u e6
the khai lri2n lhanh mQt ehu6i hQi t~1dell tren vung e~n nay.

,/'
dWi/(

. I.

'-/;J

~:i;/'

';/(;;jt.'c5ao

(j~9c

49

:!(/(J/

'

CD

j~;?JfZ'lt- c?hCVlth~i

00

HeX) LPm (x - a) = L Um
=
(x)
m=O

m=O

Lrung d6 moi UIIlla da lh(tc di€u boa .
Do KlulIll(x)= Ix12-1I1I1Il~2

[ Ixl J

va chlloi da lh(tc

I

Um

hOil~l d€u

III =0

w

Lren Va ,La suy ra c1L((,ic hlloi
c

L

Klull1} hOi l~ld€u lrung mOl Hin c~n clla b,

=0

111

khid6

,~KllIm}~{~Um
]~ Kill]
Ham s6 Klulla tC1ng mQLchlloi cae ham K[ulll},ma Cell: am
cua
h
KllIlIl}dicu boa lren E*c R'\{O} (kel qua bl(OCI), nen lheo dinh Iy II.6
(chuoung4) Lac6 Klu J la h~\Jndi€u boa lren R1\ {O}.
IhiO'c 3 :
Gj~1Sll'KllllIA ham diell hoa ,do u=KIKlulJ nen then ket qui1clla buoc
2 La c6 1I I,} hAm c1i€lI hoa

,

LJ
CJlIi lhich:
.

MQLpluwng plu-1pkhac d~ chang minh dinh Iy Lren la dlfa VaGbi~u

th('o K[u I(x) ~

[x12 -n

1I(

[X~2)

di! tjnh cae l1~n ham dong philn cua K[u] then

cac d~lo h~\Jl1
rieng phgn Clla u,sau d6 tinh i'1K[u] va chang minh bj~u th((c
nay bhng (),

[V-NGHI(~M CUA HAl TOAN DIRICHLET MIEN NGOAI.
lIBjllh Ii :
Xet bAi Lm1n
Dirichlet d6i voi mien ngoai coa qU(1 don vi B trang RI1
diu
(vdj n> I)
i'1u(x) = 0, '\Ix E R"\ B

HeX) rex),'\Ix E oB
=
Lrongl1{)r lien t~lC
tren bien oB va nghi~m u E C2 (R"\ B) n C (R'\B).
MOLnghi~m d~c bi~Lella bai loan nay la


(/'
j.fUj/{

,y'
(£'
f/ rl/~ 6ao

j f

lxl2

~.

U(X)=

w

al3

1

I

-

x

~iI."
cree

so

l!OO/

I

-

II

.jfj.;:y&~ 5YhMA %

neu

f(~)dS(~)

xERII\B

~
I

neu

t"(x)

x EoB

lrung (16w la di~n lfch ciia m~l du don vi .
C/lllllg

B nht(

XCl ham v tren

~
v(x)=

f

sau

l-1xI2.

I W aBlx -

l1linh

nell x E B

~r t(~)dS(~)
nell x E oB

rex)

Theo c()l1gthCrcPoisson lhi ham vEC2(B)nC( B) va v dj@lIhoa tfen B.
Ham v xac djnh lfen B\lO} thl ham Klvi xac djnh tren R'\B.
Theo L1jnhIy llI.6-cht(dng 4 ,K[v] co bi€u th(rc sall

A'

Klul(x)=

I

0)

f!xl2

II=I

- I

Ix-111

neu

x E RI1 \ B

f()dS()

rex)

nell

Ixl = 1

{--HlIll di@uhoa tfen t~p md B\ {O}nen theo djnh Iy 111.7
v
-chuang 4 la
c6 Kl v Il~1 h~lIn di@u hoa lfen R"\ B .
-D€ chang minh KI u I lien l~lc lfen R"\B, ta chI dn
cib Klvll<.li 013.
Cui

~ lllY 5' lhuQc oB

Neu x -7

~ trong

R"\B

.
lhi

x':::; -;- --j- ~:::;
IxlI~I

~

Jim Klv](x)=
x -7 ~

lim 11
x

X -7 ~

:::;~,

hk do v(x*)-7f(~).

I

2-11

Do LV)

ki€m tinh lien t~IC

X

v

---;;

=f(~),

( Ix/- J

v ~y KI v I lien t~ICtren I~"'\B.
Ham sf{ Klvl thoa cae tlnh cha't ciia b[ti loan nen la mQI nghi~m eila b[li
loan.
0


,5.£:(2/1 CPa~~«(feLO c1l~c

51

!!!{/o/

,/t:Y~t;!I&~ ~9'h(Mih %

2/ Dillh Ii :
Xet bAiloan Dirichlet doi vdi mi~n ngoAi clla qua du dejn V!B trong Rl1
(vdi n> 1)
L1U(x)= 0, \:Ix E RII\ B
HeX) ::: rex), \:Ix E 3B

trong d6 r lien l~IC
lren bien 3B.
Khi d6,mqi nghi~m
lren nll\ B nilL(sau

x

:::
I

1

X

2--11

u(x)

w ~

l

uEC\R"\B )nC(R'\B)

+-

[ Ixr ]

clla bAi loan d~u c6 d<.tng

)

x--l

III

r(~)dS(~)

, \:IXE n!\B

CD
3131x ~111
-

trong de):
w Iii l11e)th~1111 di~u boa lren B\ {O},lien t~ICtren B \ {O} va c6 gi::;6
tren bien 813 b~ng O.
CDI~Idi~n llch clla lI1~l call (jdn vi.

CJul tbieb:
.Trong l\"L(Ong
h(}p n=2 thl U co lh~ Vie'l dl(oi di:.\I1g
san

I
u(1',H)

1

= w(-,8)+-

2"

J

l'

1'2-1
?

2rc 01'-

f(p)d(p
+ 1- 21'cos(8 - (p)

, Vel',8) E (1,

CX)

)x(O,2rc)

CluIng millh
'"

J
-Gqi

h(x)=

CD
1I:::llx -

rex)

f()dS()

neB

x E n.11 \ B

111
'"

neB

Ixl:::1

Ta lh{{yh IA II1Qlnghi~m clla bAi tOi:ln(lheo djnh 19 IV.l d lren).
Gia Sl( II lA mQt nghi~m ba't ky cua bili loan dang xet.
f)?l g(x)=u(x)-h(x)
Ta Cl) g Iii mOl ham di~lI boa tren R"\ B ,lien 19c t1'enR"\B ,c6 gia td 0
lren bien aB.


(?

.L(trilt

.'j,'

(a

c".'

52

F Ii", lDaD
?(9C 200/

Jt;~~JJt- 3hlVlb~%

(:;l)i w=Klgllhl w X,1c((jnh trcn B.
I-Hling di~ll boa tn~n t~p mo' R"\B nen w la ham di~ll boa tfen B (do
dinh Iy IIl.7-cllltdng 4).Ham g b~ng 0 tfen aB nen ham W cung b~ng 0 tren

as.
-Coi

~ ILlY Y thllC)C aB

Nell x -+ ~ tIll x' = -;- --7-; = ~ = ~,d6ng thdi g(~)=O,
I

x 1-

I~I

1-

2-n

do d6

lim w(x)
x -+

~

=

lim 11
x

x -+

~

X

g ~

= 0,

( Ixl- )

SllY ra w lien t~ICt~li ~ .
Do d6 w lien t~ICtren B.
NI1lev~Y w th6a cae Hnh cha't Hell trang dinh Iy.
-Ta Cl) w=Klgl nen g=Klwl(theo

+ hex)

ll(X) :=:KI wJ(x)
2-11

= 11
x

dinh Iy llI.5-chlWng 4) ,SHYfa

w

I

\
Ixl

1
+

CJ.)

112

1

, '\JXE RII\ B

II x - n f( ~)dS(~)

aBlx-~1

0


.Yi~'~ht 'l(i~l (am (jac

53

200/

.-,A:f~~!lJJb
.9h{UJt~%

3) Vi d{l:
Xcl bAi loan Dirichlel d6i vdi mi€n ngoai ciia dla troll don vi B trang u?
') -

= 0, '\Ix E R-\ B
= rex), '\Ix E 8B

L\u(x)

(i)

u(x)

(ii)

HeX)== L

lim

(hfj'uh~ln )

(iii)

IXI-7oo

lrung dt') r lien [~IC
tren bien 8B va nghi~m llEC2(U?\ B )nC(R\B).
tHy chang rninh kef qll(1sall:
-Tnrong h(}p L= 21Itff(l;)LlS(l;;) :

an

Nghi~m clla bfti to<1nla Lluynha't va co bi611thuc tren R2\ B lEi
ll(X)

-Tn()ng

IH}PL:;t:-f_It

==

~

-I
f lxl2 ~12f(~)dS(~)
2It aBlx -

ff(~)JS(O
an

BAi loan lren vo nghi~l1l.

Cllllllg mill"
Gia sLfbai loan co nghi~m la u.

Th~o c1jl1h IV.2-chlfc1ng4 ,nghi~m u co d<;lng
Iy
u(x)=w(x *)+h(x)

~
v(1i h(x)=

2It

f

lxl2_1

anI

x - r

2 f(~)dS(~)

lIeu

?

xER-\B

-

l
'-:>

neu

rex)

x E8B

')

Suy ra w(x*)=u(x)-h(x)
Dod6

w(x)=u(x*)-h(x*)

, '\Ix E R-\B.
, '\Ix E B\{O}.

Khi clH> X-7() trong B thl X*-7oo , u(x*) -7L (gi~l thief) va


.

(:I<;~ldL
'f(?~{ '(,;~o ~c

h(x"') ~.~

54

200/

ff(~)us(~)

,jJ:j; 'j/b/L '%V/btf

%

(tinh chftt clia cong thltc Poisson)

211:
aB

Do lit> khi x~O trung B thl
1

f
au

w(x)-->I.J -211:

'

(ta gqi gidi h',111
nay 1£1 )
L'

l(~)dS(C;;)

-Sau day ta se dllYng minh w bang 0 tren B\{O},
Xct ham g,lx)=w(x)+E(-lnl xl)
IHlIl1gt;ui~u boa tren B\{O}.
Khi X~O lhl gix) ~oo, suy ra g,,(x»O nell x (hi gan O.
l-)~t m=inf'{ g,,(x) :xEB\{O}} (m c6 thi la -.00).
T6n t~liday{adlrong

B\{O}sao cho gt;(ak)~m khi k~oo.

Day {ad Ham trong t~p compaGt B nen t6n t<;1i con {bdhQi
day
tu v6 hE B
,

Neu hEB\{Ollhl ham di~u hoa g" d(,lt gia tr! clfe tiiu khi x=b,
lhcn nguyen Iy Cl,retieu lhl g" 1£1ham hang (luon bang m) tren
B\{0 I,ma khi x gan 0 thl g,,(x»O nen suy ra m >O.Neli bE aB thl m=O
dn go;{h)=O
.Neu h=O thl III;?0 do g,,(x»O khi x gaB O.Do d6 ta luon c6
III ;:::O,suy ra g,,(x) khong am tren B\{O} vdi mQi E >0, do d6 w(x)
khtHlg [1mtren B\{OI,
Trong biiu lluk cua gix) LaxcL ham di~lI boa -w thay VIw,ly
lu~11lu'dng H,rla c6 ket qua -w(x) khong am trenB\{O Lt(rc fa w(x)
khol1g llLrdngLren B\{OI,

Ta ua ch(rng Illinh w(x) VITakhong am VITakhong dlrOng tren
B\{O},l1cn C()the suy ra w(x) bang 0 Lren B\{O}.
Dn (It>va bi6u th((c clia u LrenU,z\B la
u(x)

= w(x*)+h(x)

= hex)
')

=~
211:

j
au
.

l

-

lxl- - ~ f(C;;)uS(C;;)
x-c;; I

-Ham s6 u c6 biill th(rc d Ln3n th~t slf la nghi~m clla bai loan nell u
Ihl1a de linh chftL (i),(ii),(iii).Ta Lhfty 1Ithda (i)va (ii),cho x~oo Lhl
u(x) ~~

f

f(~)dS(~)
211:
an


(j.)

( ",'

'/'

oLU-{?-/t Frz,/t

c.~)

55

C)(tu en t?c !to{N

Jt:;~<;yb;?M(Mt~%
!

do (() II chi Ih6a lInh chelL(iii) nc\, vi, chi nc'u
L==

~

2n

fr(~)dS(~)

an

V~y Lacc>keL qu~i dn chung minh.

0
4) Vi d{l
XeL b[1i L(HinDirichlcL doi v(H mi~n ngoai clia dla Lron drin vi B trong

n?

,

'-,-

')

,[.,II(X)

,11(x)

==

0, \Ix E U-\ B

rex), \Ix E 013

==

.11bi ch~n d vung vo

Cl,fC,

ILtcIii :

3M>O,3 [>0, \Ix E U2\13,1 >1'~ I HeX) xl
I

lrong de) r licn LI.le
trcn bicn 013vii ghi~m UEC2(U2\B)nC(n?\B),
Hay eh(fng minh nghi~m clia biii loan Jii
')

I
l1(X)

==
---

2IT

'x'-

fI

-

an x -~

J,

?
- l(~)dS(~)
I

Cldtllg millh
Thco djnh 19 JV.2-chlwng 4 , nghi~m

U c6 d<:1ng

U(X)==w(x*)+h(x)

?

vdi h(x)==

f~
2

"

nell

Ixl- - J f(~)dS(~)
aLlx
-~12

,

1 rex)

IH~ 11

".

Suy ra
I )0 d c)

w(x *)==u(x)-h(x) , \Ix

E

?

XEU-\B

-

x EoB

nhB.

w(x)==u(x*)-h(x*) , \Ix E B \{O}.

Khi cho X-70 Lh. X*-7W,
hie de) h(x*) -7

~ fr(~)dS(~) (1lnh chat ctia c6ng Lhue Poisson),
2n
an

(Icing IhlJi lu(x *) 1< M khi I x*1 dti /dn (x*

E

n?\B),


1£'

.'"

j{{~i/{

'

( '(tv

Yrt/t'

do d6

(k/~

JL~C

56

j!{j() /

I w(x) I hi d}~n hdi

M+I~

J1:;;ljIblt 5YicWthtfi

ff(~)dS(~)1
DB

khi x gan 0,

XCLha m g,lx)= w(x) -/-1=;(
-Inlxl) .
Ham g"t,li6u boa Iren B\{O},
Khi x~() ,Ihl g,,(x) ~CX),SHYra g,,(x»O ntll x dLi gftn O.
f)[,il m=inf{ g,,(x) :xEB\{OI} (m c6 Iht3Ia
'I'(inL"li day{adLrong

B\{Olsao

cho

-CX).

g,,(ak)~m

khi k~CX),

Day {ad nam Imng I~p compacL 13 Ben tan t'.li day con {hdhQi t~1v€
hEB.

Ncli hEB\{O}lhl ham di€u hoa g~ d<;ll gia Iri Cl,fCtit3l1 khi x=h ,tbeo
nguyen Iy cl,fCLieu Lhlg~ la ham hang (luan bang m) lren B\{0 },ma khi x
gin 0 Lhl g,,(x»O nen SHYra m >O,N6u hE 8B thlm=O do g~(b)=O ,Ne'u b=O
Lhl m ;:::() do gjx»O khi x gan O.l)o dt) La luan c6 m ;:::O,suy ra g,,(x) khang
al1llren

H\{OI vdi mqi E >0, do d6 w(x) khang am lfen B\{O},
Trung hit3u Iinte cLia g~(x) ta xet ham di€u boa -w thay VI w,ly lu~n

ll(ling llf Ll c() kc't L/u.i -w(x)

khong am IrenB\{ 0 },t(rc Iii 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 !,ncn c() the SHY w(x) bang 0 tren B\{O},
ra
T6m l'.li, nghi~11lclla bai loan Iren Iii dllY nhat va c6 bit3u thl'i'c tTen
n2\ B la
u(x) =w(x*)+h(x)

= hex)

=~
2rc

lxl2-1

fIx -(

DB

2 f(~)ds(~)
1
":>

0

5) Vi d{l:
Xcl hai [min Dirichkt

d6i vdi l1li€n ngoai clla dla Iron dl1n vi B trang

n2,
~

-

= 0, \Ix E n.-\ B
= rex), \Ix E 8B

,lHl( x)

,u(x)

u(x)
~ 0
. -111X
I

I

.

kill x~

CX)


( /J
,La~i/t

,"
frl/t

i /J
()(to

,.ij';;
,;1(9(;

I'
- 6.cy/,./ 'jI-a'pc-'ll..:7h£Mth j/ /;;

I

57

5!(){i/

trong U() r lien l~IClren bien aB va nghit%m UE C2 (R2\B)

n C (U2\B).

IHy clurng l11inh nghi~111 CUi.!bai loan la
')

1
u(x)

1xl--]

= -211: J-2f(~)dS(~)

~

3B x I

I

Cluj thieh :Vi cl~l3 (Iu(x) I bi ch~n khi Ix I kha 16n )la lru'ong h(.jp d~c
bi~l CUi.! i cll.1 (I u(x) Ico th6 lien ra 00 khi x.~ oo).Chung toi v~n tdnh bay vi
v
4
cl~\3 vll11116n phan bit%lr5 tnc()ng hc;iPu bi ch~n va khong bi ch~n .

CluIng lIlillh
Gd Sl~btli locin co nhgit%1111a
1I.

Thel> dinh Iy IV .2-chlcung 4 , nghit%m l\ c6 d<;tng
u(x)= w(x*)+h(x)

~
vdi h(x)=

lx12_1

nell

J

xER2\B

211: Blx- ~12- (~)dS(~)
3
r
nell

rex)

x EaB

?

Suy ra

w(x*)=u(x)-h(x)

Do c1()

, \: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..-

Jr(~)dS(~)

(tinh cha't cua cong th(tc Poisson),

211:
3B

Xet ham gJx)=

w(x)+£(-lnlxl)

,vdi XE B\{O}.

Ham gt;di~lI hoa tren B\{ O}.
Ti.! c6
gE (x)

= u~x*) -

h(x*) + Einlx *1

lI(X*) - h(x*)

=:

(

Khi X~O thl

J

Inlx *1
I

x* ~oo,
I

u

+ £ Inlx -I-I
gE(X)~oo, SHYra gt;(x»O nell x du gfin O.

D~t l1l=inf{ gt;(x) :xEB\{O}}
T6n l<.liJay{adtrong

(m c6 th61a

B\{O}sao

cho

-00).

gt;(ak)~m

khi k~oo.


(£'

<'.'

j{a}lt

"

':/#'

Frt-It 6au

{/'['!c

ss

200/

Day {ad nam lmng l~p compact
hE B.

Ne'u hEB\{O}lhl

JfJt~t;jfblt <91{t;ltk c;{i

B nen l6n l~i day con {bk}hQi l~1v€

h~lll1 di~u hoa gt; d',ll gill If! Cl,tCtitSu khi x=b ,tIleD

nguye n 19 clfc li6u Lhl gt; la ham hang (Iuon bang 111)Lren B\{ 0 },ma khi x
gan () Ihl gjx»O nen suy ra m >O,N6u bE aB thl m=O do gt;(b)=0,N6u b=O
Ihl m 20 do gt:(x»O khi x gfin O,Do d6 ta luon c6 111 O,suy ra gE(X) khong
2
am Lren B\{O} vdi mqi E >0, do d6 w(x) khong am Lren B\{O},
Twng
Il(dng It,r la
Lren Ii \ {° I.

hi6u LhC(cclla g,,(x) la xet h~lIn di€u boa -w thay VI w,ly lu~n

kCI qua -w(x) khong am trenI3\{O},tC(cIii w(x) khong dl(Ong

Cl)

Ta da cIlll'ng minh w(x) vua khong am vua khong dlWng tren
B\{O},ncn c() Ih6 SHY w(x) bang 0 lren B\{O}.
ra
Tl)l111',li,nghi~m dla bili Loan tren Iii duy nha't va c6 bitSu tllll'Clren

n?\ B Iii
u(x) =w(x*)+h(x)

= hex)
?

=~
2IT

f 'X'- -~ r(QdS(~)
x-~

aB
I

I

0

6) \Ii d{l:

,

XCI hili loan Dirichlet l!()i vdi mi€n ngoiii clla dJa Lron ddn vi B trong

n- .
")-

,L\u(x) = 0, '\Ix E R-\ B

(i)

,u(x) = rex), '\Ix E aB

(ii)

.

u(x)
_

II

~

L:;t)(

"'

'

kl11 x~

CD ( III )

III X

Lrung do r lien ll,lCtren bien aB , L c6 thtS bang +CDhay -CD,
Vtl nghi0m

UE C2 (R2\

B)

n C (Ie\B),

Hay chCrng minh ke'lquLl saIl :

.

Khi L hun tH~nIh1nghi~ m Ii}<.Iuy
nha't va c6 bi€u Ihuc tren n?\ B la
I
II(X)= L .Inlxl+-

lxl2-

2IT

f

J

-~r(c;,)ds(c;,)
-

1
DBIx - c;,


. L(;ri/l

'l(z~(

.

0(;0

,~(;

Khi L h~ng

+0':)

A

S9

':;(J(J/

hay

-UJ

/'

~.

'j/(t;7C/t

()//
cp;'
,j !z{{//tlt j/ it

Lhlbai tOLln nghi~m,
va
CIl {tllg Illillh

Gj,i sil' h~\i Loan lren c(} nghi~m

Iii lI.

Theo dinh 19 IY,2-chLwng 4 , nghi~m II c6 ui:,lng
lI(X)= w(x*)+h(x)
?

[ ~2n

vl)i h(x)=

f Ix/-~
x _t;;

2 f(t;;)uS(t;;)

DB
i

n~u

xER2\B

1

n~u

1 rex)

X E 813

')

Suy ra

w(x *)=lI(x)-h(x)

Do d6

, '\Ix E n.-\B.

w(x)=u(x*)-h(x*)

,'\IXE

Khi cho x-}O thl x*-}O':), u(x*)
h(x*) -}

~ ff(~)JS(t;;)
2n

B\{O},

lien wi +0':)hay -0':),

(Hnh chill clla cang Lhuc Poisson),

DB

llic li() w(x) lien Ldi +0':)hay -0':) ,

Nhu' the ,l6n l,.li 1'>0 sao clIo w lllan Iuan dL((Jnghay Illan Iuan am tren
!

mi~n 13(0,1').M:;tt klulc, hiim s6 di~u hoa w tri~t Lieu tren bien ClB.
Theo dinh 19 Il.3-chlcdng 4 ,ham W c6 bi€ll thue tren B\{O} Hi

w(x)=k(-Inlxl)

.

'\IxeB\{O}

Ll1c <-16 c() bi€u Lhtre [ren n?\ B Iii
u
.

u(x)= w(x*)+h(x)

=k(-lnlx *I)+h(x)
= k.lnlx/+h(x)
H~lIl1S() u c(} bieu thlk d [ren Lh~t sIr la nghi~m clla bai LOan nell u
lh6a dc

linh chilL (i),(ii),(iii),Ta
u(x)

thily u lh6a (i)va (ii),cho X-}O':)till

-} k

Inlx!
do (I() u lh()a Linh chill (iii) n~u va chi neu
L=k

va L hull h<;ln


.~L;(i/'

60

frt~1 (>{~O
i;t~(; !!()()/

V~y khi L hang

.A:f;t;YIJ/t !JZ,Wtli%

+lfJ hay -0) lhi hai lOLin v() nghi~lll,con

khi L hecn h<;ln

lhi nghie III u la tllIY nhal va co hi€lI lluk tren c() bi€u lh((c Iren U?\ B

la

')

I
u(x)=

L .Inlxl+--

2n

ix 1- - j

f x-c-;; ') f(~)ds(~)
~

.

DB
I

I

0

7) Vi d{l:
XCI hai loan Dirichlel dC)i vdi mi~n ngoai cila quit cau d(Jn vi B lrang
R" (V(1i n>2)

L1U(x) 0, '\Ix E J{I\ B
=

(i)

lI(X)= rex), '\Ix E oB
( ii)
lim HeX) L
=
(huu h<;ln)(iii)
Ixl--+0)
lrong (It) r lien Il.IClren bien oB va nghi~m UE C\}t'\B)

n C (RlI\B).

Hay chll'ng minh nghi4111 cila bai loan la tllIY nhat va co bi€u lhuc tren
}t"\ 13 Illll( S311
')

I
II(X)~

[L -

1

7-11

(0 ff(Ob(~)
aB

)

(1-lxl-

CIUl/lg

x ~-1

)+ (0 flx'OBI

n r(~)dS(~)

c-;;I

Illillh

GiJ sil' h~li loan c() nghi~m Iii u.
Thco tlinh Iy IV-I ,nghi~lllll

c() tI<;lng

u(x)= Ixr~-ll w(x*)+h(x)

r

v,1i h(x)~ 1

~

f

lxl2 - I .

nell

aBlx - 1;,1"
I(1;,)dS(1;,)
rex)

X E nil \ B

(()

l

Suy ra w(x*)= Ixlll-2 (u(x)-h(x»
Do J() w(x)= Ixlz-n (lI(x*)-h(x*»

nell

, '\Ix E R"\B
, '\Ix E B\{O}

x EoB