1. In this paper we construct the Green function and the solution of the equation

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: P RACE MATEMAT Y CZNE X IX (1976)

M. F

(Krakow)

O n a certain boundary problem for the equation (A + a2) ( A + b 2) u = 0

1. In this paper we construct the Green function and the solution of the equation

(1) {A + é ) {А + Ъ2)и{Х) = О

{ X = (aq, ..., <jcn) and а, Ъ are positive constants, а Ф b) for the domain E+ — {X: аа{€ ( — <x>, oo) for % = 1, 2, ..., n — 1, œn > 0} under the boundary conditions

и{$ог, ..., ®n- \ ) 0) = fi{$ii • • • ? œ„-i) ?

( 2 )

Au(ûff1} ..., 0) = / 2 (^ 1 ) • • •} 1 )•

2. We give now some formulae, which will be needed later. Let X and Y be two different points in n -dimensional Euclidean space E n {n > 2).

Let

П

r = = [ î > « - ÿ < ) s]ie i=l

be the distance of these points. We now consider th e function (3) U( X, Y) = U(r) = - a - 2v{br)-vY v(br) + b-2v(ar)-vY v{ar),

where v = (n — 2)/2 and Y v(z) is the r-order Bessel function of the second kind. Since U(r) is a linear combination of solutions (ar)~vY v(ar) and (br)~vY v(br) of the equations

{ А + а г) и { Х ) = 0 and {A + b2)u{X) = 0,

respectively and thus U(r) as a function of X (or Y) satisfies equation (1).

If we use the formula ([2], p. I l l )

and the expansion of Y v(z) into series ([2], p. 110 and 113) we easily obtain

(4) U(r) — | 0(r~2v)

o(r~x)

for v > 0, for v = 0,

(5) dU(r)

---= a z b(br)

dr ~vY v+i(br) —ab~2v(ar)~vY v+l(ar)

| 0 ( r - 2”+1 for v > 0, - |0 < i) for v == 0, (6) AU(r) + (a2jrb2) U(r) = 1 i

~'Y,(br) + l>‘-*'{ar)-T ,(ar) 0(r~2v) for v > 0,

o(r~l) for v — 0, (7) 4 - [ A U ( r ) + (a2 + b2)U(r)]

dr

= ba2~2v (br)~v Y v+1(br) — ab2~2v (ar)~v Y v+1(ar)

= 2r+1 r (v + 1 ) n - 4 - 2va -2v{b2 - a2)r~2v~l + H v{r), where

H v(r) = 0(r~2v+1 0

1

for v > 0, for v = 0 (the symbol 0(f(r)) (o(f(r))) refers to the case r-> 0).

Let D be a boundary domain whose boundary we denote by dD.

Let dD be of the class Gf Let w, v be functions which are of class Gi in D and of class G3 in D v d D . Using Green’s formula ([1], p. 230) to the system of two functions v and Aw, Av and w, (a2 + b2)v and (a2 + b2)w, respectively, and adding obtained equations, we get

(8) j {w[A2v + (a2jrb2)Av] — v[A2w Jr(a2-\-b2)Aw]}dD

D

— f !v-^-[Aw + (a2 + b2)w] — - ^ - [ A w + (a2 + b2)w] + A w ^ — —^ - v \ d S ,

J dn dn dn dn )

dD 1

where n is the inward normal to dD.

T

1. Let и be a function of class G* in D and of class C3 in D v d D satisfying equation (1) in D. Then

1

(9)

V n ^{dD y}

/I и - Л - [ А и + (а2-{-Ь2) Щ - — [AU + (a2 + b2)U] +

dnA dn

„ dU + Au ——

d'il jr

dAu dn

u(X) 0

for X * D \ d D ,

for XeC( D\ j dD) ,

where yn = Qn2v+1 Г ( у 1(ab) 2v(a2 — b2) and Qn is the surface of the n-dimensional unit sphere; G(DudD) denotes complement of D u dD relative to E m.

P ro o f. Let Xe G( Du d D) . Applying (8) to the function u ( Y) and U(X, Y) as a function of Y we get (9). Let X be interior point of the domain D and K R denote the ball with the centre X and radius R, K R c D . If in formula (8) D is replaced by D \ K R and functions w and v are replaced by u ( Y) and U(X, Y), respectively, we get

r i d dU

(10) \U —— [Au + (azX b z) u1 --- -— \_Аи-\-(аг-\-Ъг)и\ +

^ J ( (л/Yb dfly

dD

л du dAU ) + AU —--- ---u\dSy

dn dn y J

dK

U [ d P + (a2 + &2) Щ du

dn [dZ7 + (<*2 + 62) U] + Au dU dAu

dni dn u ) d S y . Applying the mean value theorem to the surface integrals we obtain

d

r- = ^dKT? / ^{u dn,} [AU + {a* + b*) U]dSY

anBn- ' u(Qt) -[z)P (r ) + (e> + bv U(r)]Us, d

dKr> f

du

d n T7

[AU + (a* + b2) щ а в 2

, du(Qo)

= QnRn M Щг) + («2+*>2) Щг)3U n,

Z3 = j'Au dU dSY = QnBn- l Au{Qz)^-V{r)\r=R,

dK

dn dr

R

- f

dAu , dAuiQ/,)

TJ ~ — dSy = QnBn --- — U(R),

dn dn

where Q{e dKR (i — 1, 2, 3, 4). Taking into account (4), (5), (6) and (7) we obtain

h = Q bBT-'uU iJ i2v+lr{v + ±)K-l (ab)-2v(b2- a 2) R - 2v- l + H v(B)-]

(* = 2, 3, 4),

hence by (10) we obtain (9).

3. We now pass to the construction of the Green function of equation .(1) and for the set E + . Let X and Y be two different points, X e E + ,

Y e E + . Let us denote by X = (aq, ..., —æn) the symmetric image of the point X with respect to the coordinate hyperplane yn — 0. Let us write

71

71 —

q = 1 Г = ( ® < - y < ) 2 + (®» + y » )2] 1/2>

Q = [ £ ( V i - V i f + x l ] 1'2'

i= 1 i= 1

We are going to prove T

2. The function

'(

11

G(X, Y) = U ( r ) - U ( r x),

where XJ(r) is given by (3) is a Green function with the pole X for equation (1) and for E f with the boundary conditions

'(12) G(X, Г )|Уя_0 = 0, Ar G(X, Y ) |%=0 = 0.

P ro o f. U(rx) as a function of Y is defined and of class C°° in E+

and satisfies equation (1) in this set. Por Y

dE+ we have r = rx

q

•and G{X, Y)\yn=0 = U( q ) — U( q ) = 0 . Since AY U(r) = V"{r) + r~l {n- - l ) V' {r)

and so Ar G( X, Y ) |%=0 = [AT U(r) - Ar TJ{rx)% n^ = 0 .

4. Let u( X) be a function of class (74 in E f satisfying equation (1)

;and the boundary conditions (2). Applying formally formula (9) to func­

tions U(r) and — U{rx) and for В = E+ and adding obtained equations, we get, in view of (2), (11) and (12),

•(13) u(X) = — f \ м Т ) - £ - [ А у 9 ( Х , У ) + (ал + Ъ*)в(Х, Г)] +

У п v J i д У п

д 11

+ М Т ) — в ( Х , Y)

дуп 1 d Y ’,

Vn -

where Y ' = {yx, ..., yn_x) denotes a point in (n — 1)-dimensional Euclidean -space Е п_г. I t is easily shown, using formulae (6) and (7), th at

' (

11

d dyn

d

дУп ? )\Vn=o 0пя я(е),

[Ar G( X, Y) + (a* + b*)G(X, Y)]|%=0 = x nX x(6),

N ^ q ) = 2b2- 2va2(aer v~1Y r+1(aQ )-2a2- 2vb2(ber v- 1 Y v+1(b6), N 2( q ) = - 2 b 2a - 2v(bQ)-v~1Y v+1(bQ) +2 a 2b-2v(aQr v- 1Y v+1(aQ).

w h e r e

Substituting expressions (14) into (13) we obtain

X г

<15) u { X ) = - ^ - [ /1( Г /)ДГ1(е) + Л ( Г ) ^ а(Р)]й Г .

У п „ J __

1

L

1. Lei \fi(Y')\ ( i = 1, 2) be a function which is Lebesgue inté­

grable over E n_1. Then the function u(X) given by formula (15) is of class C°°

in E f and satisfies equation (1) in this set.

P ro o f. We shall prove th a t the integral on the right-hand side of formula (15) as well as the integrals we get by differentiating /г-times (/г = 1, 2, 3, ...) with respect to xi (i = 1, 2, ..., n) the functions under the sign of integral (15) exist and are uniformly convergent for X belonging to the set

Q = {X: \Xi\ < 0 , i = 1, 2, ..., n — 1, A < xn < B ) ,

where A, B G are arbitrary positive constants. The functions we obtain by /г-times differentiation of the kernels N x( q ) and N 2( q ) with respect to xt (i = 1, 2, ..., n) are linear combinations of the functions

71

ф = ^{[ J}

П : 1

where £ аг- < // and /9 > 0, a{ > 0 (i = 1, ..., n — 1), h = a, b. I t is enough to know th a t the integrals ill

(16) J f t( T ) 0 d Y ' (i = 1 ,2 )

E n- 1

exist and are uniformly convergent for Xe Q. For X e Q we have q ^> A.

N

from the assymptotic properties of Bessel functions ([2], p. 145) we see th a t Y v(hg) are bounded for q > A. From these results wre can obtain the inequalities

m < M av an^

for X e Q , Y e E n_Y, where M a f0lntp are positive constants. I t follows fiom this th a t

I / / , ( г ' ) « 1 ' к ж ( En— 1

I Ш Г ) \ Л Г

E n - 1

(i = 1 , 2 ) .

Hence and by assumption of Lemma 1 integrals (16) exist and are

uniformly convergent for Xe Q.

The function u( X) defined by formula (15) or (13) is of class C00 in the domain E£ and its derivatives may be found by differentiation under the sign of the integral.

We shall prove now th a t the function defined by formula (13) satisfies equation (1). Taking into consideration the above properties and the fact th a t the function G(X, Y) as a function of the point of X, X Ф Y satisfies equation (1), we have

L

2. Let |/ ( Y ' ) | be a function which is Lebesgue integrable over Е п_г. Let /( Y') be continuous at the poit the X'0 — (#$, ..., Then the function

is defined for X e E+ and convergent to f(X'0) when X~>(X'0, 0-f).

P ro o f. If we use the formula

{Qn is the surface of the ^-dimensional unit sphere, n > 2) we easily find th a t [3]

+ h ( T ) (а? + Ьг) Х ( А х + а?)(Лх + Ъг) в { Х , Y) + dyn

(17)

Multiplying both sides of (17) by f(X'Q) we obtain

We now present the function L ( X) in the form

w h e r e

2$ f*

° {X)==~Q~ J U ( ^ ) - f ( ^ ô ) ] Q - ndY'.

I t is enough to show th a t C(X) is convergent to zero if X-+(X'Q, O-f).

Let s > 0 be given. I t follows from the continuity of the function /( Y') a t the point X'0 th a t there exists the (n — l)-dimensional ball K R(X'0) with the centre X'0 and radius В > 0 such th at

<18) If ( Y ' ) - f ( X ' 0)\< 812 for Т е K R(X').

Let

{19) G(X) = Сг(Х) + С9(Х),

where

2cc г

Сг( Х ) ^ ^ [ f ( Y ' ) - f ( X ' 0) l Q- nd Y ,

^n J , Krtx0)

2x г

°2{X)==lf J U ( Y ' ) - f ( X ' ) l Q- nd T , C[KR(x'0)\

where C[KR(X'0)] denotes complement of K R(X'0) relative to Bn_l . I t follows from (17) and (18) th at

{20) |C ,(X )|< e/2 for X e E + . '

Let X ' X'0< В /2 and Y ' X ' > B. Then X ' T > Y'X'0- X ' 0X ' ^ E/2 and C[KR(X'0)\ cz C[KRI2(X')-}. Hence

{21) \Ca( X ) \ ^ C t (X) + Ct (X),

where

Cz(X) = ^ \ f ( X ' ) \ f Q~nd T , n C[KRI2(X')]

G l ( X ) = l i ï ~ / l/( y ')l n C[KRj2(X’)\

Introducing the spherical coordinates in C3(X) Vi =<»! + »* cos 9?!, y 2 — a^ + rsin^cos^g?

+У

= ®„_irsinç>1...sinç?n_2,

where JR/2 <

* < <x>, 0 < y{ < -к (i = 1. ..., n — 3), 0 < <pn_2 < 27c, |J |

= V * ~ 2 V>{ <P l, < P n - b ) l w e g e t

OO О

C 3 { X ) = M æ n

J

RI2 Rl2

Г 7T Z 1

= Ж ---arctan —— ,

L 2 2®nJ

dr r2 + <

where i f is a positive constant.

Since the last integral is convergent to zero for æn~>0+, there exists a number rj > 0 such th a t the conditions 0 < con < rj and X' X' 0< В / 2 imply

(22) C3( X ) < e / ± .

Let us now take into consideration the integral C4(X). For Y 'X ' < В /2 we have B j 2 and thus

2&„ Г

J

n С[ХД|2(Х')]

|/( Y ') |d Y '< 2a?n Jf,

where J f x is a positive constant.

From this result and by assumption of Lemma 2 it follows, th at there exists a number y 1 > 0 such th at

(23) C4( X ) < e / 4 for X' X' 0< B]2 and 0 < a?n < î? x.

I t follows from (19), (20), (21), (22) and (23) th at

\ G( X ) \ <s for X ' X ’0< B / 2 and 0 < wn < min(^, ?yx).

L

3. |/t-( Y')| (i = 1, 2) a function which is Lebesgue inté­

grable over En_1. Let fi (Y' ) be continuous at the point X'Q — (a?°, ..., a?°_x).

Then the function u(X) defined by (15) satisfies the following boundary conditions :

(24) limit(X) = f 1(X'0) as X ^ ( X

, 0 + ) , (25) lim zl u(X) = f 2{X'Q) as (X

, 0 + ) .

P r o o f . We shall now prove th a t the function u(X) defined by (15) satisfies condition (24). I t is easily shown by using the expansion of Y„+1( 0 ) into the series and the asymptotic formulae for these functions ([2], p. 145) th a t the functions Wx(^») and X 2( q ) may be written in the form

(26) X ^ q ) = 2 v+2{ab)-2vr ( v + l)7i-1Q-2v- 2{a2- b 2) + P ( e),

(27)

w h e r e

P ( q ) =

m+1) as q 0 -f- j X

( q ) =o(Q~n+1) as e ->0 + .

\

Now let us write u(X) given by (15) in the form u(X) = Jx{ X ) + J ^ X ) + J 3{X), where

J i(X) æn2

Г (v -[-

){сг — b Yn(<*)*n

j 2(X) f м г ) Р ( д ) а т ,

Yn J

f u r n Em ___ i

Q~ndY',

Ern _ 1

СО Г

J

(X) = -* - f

( T ) N

(Q)dY'.

Ym _J

Yn

JL,n— 1

In view of Lemma 2 and definition of yn we have Ит«7х(Х) = f x{Xо) when X -> (X

, 0 + ).

I t is enough to show th a t integrals J { ( X ) (i = 2, 3) are convergent to zero if X->(X'0, 0 + ). We shall show this only for the integral J

{X)r the proof for the integral J

(X) being similar. Since P ( q ) = o(g~n+1) as

@->0, thus there exists a number q

> 0 such that (28) \P( q ) \ < Q

for 0 < Q< Q0.

I t follows by the continuity of the function f x(Y') at the point th a t there exists the (n — 1)-dimensional ball K S(X'0) with the centre and radius ô > 0 and a number M * > 0 such th at

(29) Let

\ Ш ' ) \ < Х д for Y ' e K d(X'0)

e • .

Qo ^ S l = m m \ - , -

The integral J

(X) may be w ritten in the form

(30) J x(X) = B

(X) + B

(X),

where

B 1( X ) = ~ f A m ) ? ( e ) d r , k

( xq )

00 I*

B 2{ X ) = - ± M r ) P ( é ) d r .

Yn J ,

C[Kôl(XQ)]

N N

If X'

X ' < (5x/2 and then Y 'X ' > Y 'X '- X '

X ' > <5,/2 and 0 [Х в1(ХЙ] c C № l/2(X')]. Hence

л? /*

|Яа( Х ) | < - ^ |Л ( Г ) ||Р ( е ) |Д Г .

У п

C[Kôlj

(X')]

For Y ' X ' > <5,/2 we have q ~^ ô

f2 and \P{ q ) \ ^ M 2, where il/2 is a positive constant. Thus

|Б 2( Х ) |< Ж 2

У п

f Ifi(Y') ^I dY'.

Let e > 0 be given; then by assumption of Lemma 3 there exists .a number Ç > 0 such th a t the condition 0 < xn < £ implies

Therefore

Ж о

Уп l/l (Y ')|d Y '< e/2.

(31) |P 2( X ) |< e/2 for 0 < < £ and X 'X

< <5

/2.

Let us now take into consideration the integral P x(X). Let Y 'X

< (5X.

If X'X'

< ô

l

and 0 < xn < £x = min (£, <5x/2), then X ' Y '< X^ Y' -f + X 0'X '< (5x + (5x/2 =1(5, and ^ ( X j ) с Ж3<5 (X').

2 01

From this result and from formulae (28) and (29) we have

(С Г

l-®i(X)| < — l/i( Y')|

У п т т У

|P ( e ) |d Y '< — xn

У п

Mô J ^{Q- n+}

^dY'.

к з (X')

-<5i 2 1

Introducing the spherical coordinates

Vi = «i + rcosç?,, Уп- i = «n_ 1 + rsin^1'...-sin99n_2,

\J\

=. rn~

\p(cp1, . . . ,q >n_3) in the last integral, we get

j Q~n+

dY' = xnN J rn~

( r ^ æ

J~ n+

)l

d r ^ œ nN j

к з (X') о о

-д-i

2 1

dr

where X is a positive constant.

Since the last integral is convergent to zero for xn-> 0 + , there exists a number £2 > 0 such th a t the conditions

0 < xn < min(£i, C2) and X 'X '

< ô

imply

(32) \Bx{ X ) \ ^ e ! 2 .

I t follows from (30), (31) and (32) th a t

|J 2(X)| < e for 0 < œn < Cs ami X ' X

< dj/2, where Cs = min(£i, C2)-

So we have proved th a t the function defined by (15) satisfies the boundary condition (24).

We shall now prove th a t the function u(X) satisfies condition (25).

In view of formulae (14) we have

- Х [ ^ в (х , Y)]|%=0 ^ ~ [ А г в ( Х , Г)]|%=0

=

b

a~

v(bQ

Yv+l{bQ)æn -

ctb-

v{aQ)-v- l Yv+l{aQ)<fin

= [2'+a (ab)~

vr (v + 1 ) тГ1 q

(a

- b2) + W(e)]®ni where

TT(£) = o(g-n+1) when q ->0.

~ [ A x AT0 ( X , 7) + (а* + Ъ*)Ах в ( Х , Y)]|„n_0 УУп

ç\

= — - [ A

YG(X, Y) + (a

+ b

)A TG ( X , Y )]|%=0

™Уп

= - 2 a i b2- > (ag) - ’- 1 Y,+1(«e)æ„+26‘ a2- 2’(6 g )-'-1 Y,+,(&£>)*„

= - ( b a ) 2X 2(e)æn.

We have then

Au(X) =* А г( Х ) + А г(Х) + А а(Х), where

A A X )

M X )

M X )

7 n

>2

>

/ и ( Х ' ) ( - а 2Ъ*Х2(в))йХ',

2 2

vT(v + 1) {a

- b

)æri

xn

Yn

(ab)v Tzyn

J f , m V ( g ) d T '

f h ( T ) g -

’-

dY', En-X

n— 1

5 — Roczniki PTM — Prace Matematyczne XIX

In view of Lemma 2 and definition of yn we have limA

(X) = M X ' 0) when 0 + b I t is enough to show that

(33) A<{X)-+0 (i = 1 ,3 ) if X->{X'0, 0 + ) .

Now the proof th a t (33) is satisfied is similar to th at of the proof for the integral J Z{X).

From Lemmas 1 and 3 we get

T

3. By the assumptions of Lemma 3 the function u(X) defined by formula (15) is the solution of equation (1 ) in the set LJf with the boundary conditions (24) and (25).

References

[1] M. K rzy za n sk i, Bownania rozniczhowe czqstkowe rzçdu drugiego, cz. I, Warszawa 1967.

[2] N. L eb ied iew , FunTccje specjalne i ich zastosowania, Warszawa 1957.

[3] J. M u sialek, The Green's function and the solutions of the Neumann and Dirichlet

problem, Comm. Math. 16 (1972), p. 1-35.