On a certain property of the equation

ANNALES SO CLE TATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X f (1968)

J. Musi

(Kraków)

On a certain property of the equation Apu f-k u = О

In the paper [ 1 ] we dealt with the equation A%u { X ) Jr k{X )u{X ) = 0 where k(X)eC°, X = (aq, . . . , xn). We gave a condition sufficient in order that each solution of the equation vanish identically in the unit ball. In this paper we shall give analogous theorems for the equation

Д1рЫ{Х) + Щ Х)и(Х) = 0 for n — 2 and n > 3 separately.

I . Let us first deal with the case n — 2 ; z = (x, у), С = (I, rj) will denote points of the plane.

Le m m a

1 (see [3]). There exists a continuous function k(C) defined in the unit circle K x = {z: \z\ < 1 } such that the Laplace operator A applied to the function

w(z) = f f k(C)logrdC, к 1

where r = \z — £|, gives a function which is discontinuous in a dense subset of

We shall now prove the following

Le m m a

2. Let v{z)eC2p be a solution of the eguation

(1) A{p)v{z)-\- k(z)v{z) — 0

in bounded region D with boundary dD of class Cl (i.e., piecewise of class C1).

Tjet геС2р~г in D. Then v satisfies the condition

v(z) = ap y J f(S.f r2p- 2i- 2logr + Cj’r2łl- ai- 2) d r ~'~lv _

Z=0 dD L d П

— ^ ( ^ J _ <_ 1 r2llogr + C?_<_ 1 r 2 l)zf®] ds— apf f k(C)v(C)r2p' 2logrdC where A p, С?, А р^ _ г, ар, Cp_i_l are constants and ap = (2 t zAp_x)~l.

R oczn ik i PTM — P ra ce M a tem a ty czn e XI.2 19

2 9 0 J . M u s i a ł e k

Proof. Let K R denote the circle of radius В and centre P(z) con­

tained in D. Applying the fundamental formula (with the normal n di­

rected towards D, see [ 2 ], p. 18)

23-1

j j (uApv — vApu)dz-\- ^ j ( a , йАр~ * -^ dA

и dn - A lv

dn ds — 0

D i—0 dD

to the function и = r2p~2logr and to v{z) in the region B — K R we obtain (2) J 'j ' (r2p 2 logr Apv — vApr2p 2 logr)dz-\-

t 2p_2l__ dAp~'l~l v Ai^ dzP_t_V p~2logr

d- k r 23 -1

I j Axrip 2logr —---

— A lv-

j A j J \ dn dn

i= 0 dD '

j ^{ds = 0 .}

Since Al (r2p 2 logr) — A pr2p 21 2 logr-j-Cfr2p 21 2, in virtue of (1) and (2) we have

23-1

f f i(()v (C )lo g rd ( j ( A f r ^ - ^ l o g r + C f r 2”- 21- 2)

T7" _ n an L

D-KR d dn

2 3 -1

£ = 0 dD

Ap__i_xr2%\ogr A-Gp-i-i^21) A1VI ds +

dAp~x~l v dn

■]

+ j \( A ri rw- li- 4 o S r + Cfr®-24- 2) ' ’’’l ^{d s -}

i = 0 d K R P - 1

— ^ J (2iAp_i_llogr — 2iCp_i_l — A^_i_l)r21'~1 Alvds — Ap_1 J r ~ 1vds.

г = 1 d K R

dKR

Applying the mean-value theorem we obtain the following estimates for the curvilinear integrals along dKR

d-k r

^~ k {z )v {z )r 2p 2 logrdz = ^ jT(AfV2p 21 ^gr-j-O:

г = 0 dD *-

— (Avv_ i_ y '\o g r + C U - S ' ) 4 ’»1 ds +

+ V (!b z A U P - x - H o g B + Z n C f R

d r

n y . p - i - 1 (23^223-2г -2\ u

dn

г = 0 23 -1

2iAp_i_l\ o g r - 2 i C p_i_1)2nB2iAiv ( Q ) - 2 n A p_lv(Q), QedKR.

Passing to the limit as В 0 we obtain P-i

v(z) = ap ^ J \(A?r

i=0 dD *-

(.A%_i_1r24ogr + C%_i_1rlt)Alv \ ds— ap j j h(C)v{C)r2p~2logrdC

Л ^ ТЛ

lo gr + Cfr2p- 21- 2) dAp v

d dn

where av = (2-кАр~1(р — l)(p — 2 ) ... l )_1 and Л р_ ^ г, Cp_i_ x, J.?, Cf are constants, q.e.d.

From the above theorem it follows that

p

-

‘logr+C? zP_ 1 r2,>_2*_2)

г=0 dD

d dn

dAp~x~l v dn

ds — {A%_i_l Ap 1724ogr-Jr Cl_i_1Ap

ap J J l{l:)v{OAp- lr2p- 2\ogrdC.

Since Ap lr2p 2logr = A p_1logr-\-Gp_1, we obtain

p

-

(S)Ap~ 1 v(z) = a p V f UA? Ap- \ 2p~2i- 2\ogr + (% Ap- lr2p- 2i~2) r - i A L

d

dn

dn (Ap_i_!Z p-V nogr+ ds

-]

apC®-1/ / * (C)* — « г » Г Г MCMOlogrdC.

D JD

Th e o r e m 1 .

There exists a function fc(£) continuous in the circle |£|

and such that each solution o f ( 1 ) is identically equal to zero.

Proof. Let the function Tc(£) satisfy the conditions of Lemma 1.

Let v(z) be a solution of (1) of class C2P in the circle \z\ < 1 . Differentiating under the sign of the integral we see that all the terms on the right-hand side of (3) are of class C2P in the circle \z\ < 1 except, possibly, the last one which may be written as

(4) w{z) = f f h(C)v(C)logrdC

D

= v(z0) f f fc(£)log rdC + j J k (t)[ v (C )-v (z 0)]logrd£

D D

2 9 2 J . M u s i a ł e k

where z0 = (x0, y 0) and x 2- f y 2 < 1. If z0 is sufficiently close to z, we have

\ v { z ) — v {Z q )\ < C \ C - z 0\ = Gxr

where G and Gx are positive constants. Thus, the integral / / f c ( f ) O ( f ) - ® ( * 0 )]logr<*f

D

is of class G2p, because its majorant j j Gx\k{£,)\r\ogrd£, is of class G2P (see [4], p. 98). By Lemma 1, w(z) is not of class G2p, for the involved derivatives are discontinuous in a dense set 8. Since v(z)eC2p in the unit circle, Ap~1v(z) is of class G 2 for each z0e8. Thus, by (3) and (4), v{z) vanishes at each point of 8, and hence identically in the unit circle.

2. Let us now consider the ^-dimensional case. The following lemmas are analogous to Lemmas 1 and 2.

L

la (see [3]). There exists a function f { Y ) = f{ y 1, ... , yn) in

the unit ball |Y| < 1 such that the function

w{X) = S S S f ( Y ) r * - * d Y ,

\У\<1

where r = \ X — Y\

(Y(Xi—

has discontinuous partial derivatives in a set 8 dense in the unit ball.

L

2a. Let v(X) be a solution of the equation (la) Apv(X) + Tc(X)v(X) = 0 , X = (xx, . . . , x n)

of class G2P in a bounded region В whose boundary dB is of class Gl, and let v(X) be of class С2р~г in В = В о dB. Then the function v (X) satisfies the equation

p-i

= / j „ ^ S S я

г= 0 dD

,,dA Лг 2 г-п _2

--- B p v_ i _ ---A * v \ d S -

dn dn

- A . S S S * » - nh { Y ) v ( Y ) d Y

D

where B p, Bp_i_x, — ((2 — n)Bp_i_1Qn)~1 are also constants.

Proof. Let K R denote a ball with centre X = (xx, ..., xn) and radius R, contained in B. We recall a fundamental formula (see [2],) with the normal n directed inside the region:

2 5 - 1

sss (uApv - v A pu ) d Y = - Y S S U

D i —0 dD '

dA25— г —1 ,

Alv dA

dn

и dS.

dn

Substituting и = r p and v{X) in the region D — K R we get ( 2 a) SSS (r2p~nApv - v A pr2p- n) d Y +

B-KR

P - 1

ss U r 2”-"

t= 0 dD \

d A p~l~1v . d Ap- ' - l r2p- n \

Al v ---,--- \ d S = 0.

dn dn

By (la), (2a), by А1г2р~п = В р {г2р~п~21, and Ар- ^ т 2р~п = , where

B p = 2г(2р — п){р — 1)(2р — п — 2)(р — 2) ... (2p — n — 2i)(p — i) and

i>5Lt-i = 2l (2p — n){p — l)(2p — n — 2)(p — 2) ... (2i— n-{-2) (i+1)

we have

SSS л l c( Y) v{Y) dY B-KR

p - i

ZSS[B

i = 0 dB

+5'ss ^в

i —0 d K R P -1

dAp~i~1n j ^ i - 71+2

*■*»-*-« --- A 'v)d 8

dn dn

Л / |Р - г - 1 . ,

Vr2p- n+2i— --- d S - dn

- 2 SS (2*- П + 1)г«-“+1 SS (2-ti)i)^_1r1-w^ ,

г=0 З-йГд d-Кд

We have the following estimates for the curvilinear integrals

SSSr»- nh ( Y ) v ( Y ) d Y

d - kr

г=0 3-D \

vss(

,, dAp~'~l v dr2l~n+2

'р-г- i

mt*-*-* — — a ' v \as+

dn dn l*®j

+ V B p QnB 2p+2i- 1 _ 1 0

p-i

- Y B i _ i_l ( 2 i - n + 2)QnB 2iAtv ( Q ) - ( 2 - n ) D p p_1Qnv(Q),

where QedKR and Qn is the measure of the area of the unit sphere in B n.

2 9 4 J . M u s i a ł e k

Passing to the limit as R -> 0 we infer that

p-1

« д а = S S

i=0 dD

>P JLP-n—W , dA

dn

J r2i - n+ 2

^ --- A'v\ d S -

T)P . J j p - l - l

Pn SSS-№r'

D

dn

r — h { Y ) v { Y )d Y

where pn — ((2 — n)Dp_xQf) \ Hence AP £ l v{X)

p-i .

= Pn 2 s s

г=0 dD \

BfZV dA

dA ^{р—12г—п+2}

dn ^'p—i—l dn A*v\ d S -

- A . S S S Ap- 1^ p~nTc{Y)v(Y)dY.

D

Since Ap- lr2p~n = F pir2~n, where 1% = (2p - n ) ( 2 p - 2 ) ( 2 p - n - 2 ) ( 2 p - Ł) ...(2 — n)2, we obtain

(За) А р £ ^ ( Х )

= &

p

-

/

2’ ss

г=0 dD \

Р - г - i

7)P . -Up-г-!

Л /|Р-1г2г-п+2

^---- A l i S -

dn

- h ^ S S S Tc(Y)v{Y)r2~nd Y .

D

Th e o r e m l a .

There exists a function Jc(Y) continuous in the unit ball of B n, such that each solution v(X) of

of class C2P vanishes identically in the ball \X\ < 1 .

Proof. Let Tc(Y) satisfy the conditions of Lemma la and let v(X) be a solution of class C2P of equation (la) in the ball \X\ < 1. Similarly to the proof of Theorem 1 we conclude that all the terms on the right-hand side of (3a) are of class C 2P~2 in the ball \X\ ^ 1 , possibly except of the last one which may be written as

(4a) //„ S S S k { Y ) v { Y ) r 2~nd Y

D

= f n V ( A ) $ S S k ( Y ) r 2- ndY + f nS S S ^ ( Y ) [ v ( Y ) - v ( A ) y - nd Y , A e S .

D D

If A lies sufficiently near to X , then \v(Y) — v{A)\ < 0 | Y — X\ = Cr,

C being a positive constant.

The function defined by the integral

A, S S S Tc(Y) [ v { Y ) - v { A ) ] r 2- nd Y

D

is of class C2, since the function

CPn s s s \k(Y)\r3~nd Y

JD

of class C2 is a majorant of it (see [4], p. 98). By Lemma la, the integral S S S Ti{Y)r2~nd Y

D

is not of class C2p, for its second-order derivatives are discontinuous in a set 8 dense in the unit ball. Since v(X) eC2p in the unit ball Ap~1v(X) eC2.

Therefore, by (3a) and (4a), v(X) = 0 in 8, and hence v(X) = 0 in the unit ball, q.e.d.

R eferences

[1] J. M u sia łe k , On some property of solutions of the equation A^vA-lcv — 0, Prace Mat. 11 (1967), pp. 9-14.

[2] M. N ic o le s c o , Les fonction polyharmoniques, Paris 1936.

[3] H. N ie m e y e r , LoTcale und asymptotische Migenschaften der Lósungen der Helmholtzschen Schwingungsgleichung, Jahresber. Deutsch. Math. Verein. 65 (1962), p. 1-44.

[4] W. P o g o r z e ls k i, Równania całkowe i ich zastosowania, t. II, Warszawa

1 9 5 8 .