The fundamental formula for the operator (

ROCZNIK.I POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATFMATYCZNE XXIV (1984)

Ja n Gô r o w s k i, Kr y s t y n a Wa c h n ic k a

(Krakôw)

The fundamental formula for the operator (

and its application

1. In this paper we shall prove the fundamental formula for the operator ( A ±

2)p. Such a problem for the Laplace operator was investigated in [2].

These formulas we can applied for the construction of the effective solution of some boundary value problems for the equation of the type

Amu + a1 Am~ l u + ... + amu = 0, where at , a2, . . am are constant.

П

2. Let x = ( x u x 2,...,x„), у = {yu y 2, . . y„), r2 = £ (х; - у ;)2- We need some lemmas.

Le m m a 1.

I f q is an arbitrary real number, p is an arbitrary non-negative integer and c is an arbitrary positive real, then

(1) A ; ( ( c r f K q(cr)) = c 2” t ( - 1 ) * ( ^ a » ( c r ) ’ - ‘ K , _ k(cr),

k = 0 X /

к

where a0 = 1, ak = (n + 2q — 2j), к = 1, 2 ,..., p, and K q is a MacDonald

junction. j ~ 1

The simple induction proof of Lemma 1 follows from the formula

Ау и(г) = и"(г) + ^ и ' ( г )

and the formulas ([3], p. 79)

K , - , ( z ) - K , + 1 ( z ) = A ( z * K , ( z ) ) = - z - X ^ H z ) .

Similarly, we obtain

264 J. G ô r o w s k i , K. W a c h n i c k a

Lemma

2. I f q is an arbitrary real number, p is an arbitrary non-negative integer, and c is an arbitrary positive real, then

J 5 ( ( c r ) " " N , ( c r ) ) = c 2'’ £ ( “ I

y +k( f b k( c r r i°+k)N t+k(cr),

P

where b0 = 1, bk = \ \ { 2 q - n + 2j), k = l , 2 , . . . , p , and N q is a Neumann

function. j=1

By induction we have

Lemma 3.

The function (cr)s K s (cr) [(erf N _s

is the fundamental solution of the equation (A —c2f u ( x ) = 0 [(A + c 2f u ( x ) = 0], where s = p — ^n.

We shall prove

Lemma 4.

Let D be a bounded domain whose boundary we denote by S. Let S consist of a finite number of piecewise-smooth hypersurfaces. I f the functions u, v e C 2p(D) and are continuous with the derivatives up to order 2p — \ in Dyj S, then

(2) J (v(x)(A—c2y, u(x) — u{x)(A—c2f v ( x ) ) d x

D

~ X (, V — c2)p~k~ 1 £ ^(Aj u(x) — Ak~j v(x) — Aj v(x)-^-Ak~j u(x))ds,

fc=ov/c~ v j=o's dn dn

where n is the invard normal to S.

P ro o f. By the formula ([2], p. 182) J (t; (x) Ap u(x) — u (x) Ap v (x)) dx

D

= У \ ( A i u(x)-^-Ap~ 1~l v(x) — Al v(x)-—Ap~ l ~l u(x)\ds

i = o s ' dn dn

we have

J (v(x)(A —c2f u ( x ) — u(x)(A — c 2f v ( x ) ) d x

D

= У ( p S)( — c2f ~ k~ 1 ^(v(x)Ak + 1 u(x) — u( x) Ak + l v(x))dx

к = ( Л * + 1 / D

= z X âJ u { x ) ~ A k~J v{x)~ Aj v { x ) - ^ A k~j u { x) jd s ,

which ends the proof of formula (2).

Theo rem

1. Let D be a bounded domain whose boundary we denote by S. We

assume that S consists of a finite number of piecewise-smooth hyper surfaces. I f the

function u e C 2p(D) is continuous with the derivatives up to the order 2p—l in D u S and (Л—с2Уи(х) = 0 in D, then

Z Z ^ k ^ A j u ( y ) ^ A ky~j ((cr)s K s(cr))-Ajf((cr)sK s(cr)}^Ak' j u(y)Jdoy

10 for

Rn\(D и S),

I Tt"/2( — l)p_ 1 2p+n/2~ 1 (p— 1)! c2p~nu(x) for

D, where oc£ = ^ lj ( - c 2)p k \ s = p-%n.

P ro o f. We write 1^ (r) = (cr)s (cr).

(i) If x e R ”\(D и S), then r > 0 for y e D u S and Vs(r)EC2p(D и S). Hence Lemma 3 and formula (2) imply the first part of the theorem.

(ii) Let

D. We consider the ball K R c D with center x and radius R. Let SR denote the boundary K R. From Lemma 3 we get

(3) j (u (y) (Ay - c2f Vs (r) - Vs (r) (Ay - c2f и (y)) dy = 0.

d\k r

In view of (2) and (3) we can write ,

Z Z аП ( d Ju ( y ) ^ A ky- J Vs( y ) - A JyVs(y)-^-Ak- Ju(y))doy = 0;

fc = 0 j — 0 S u S R \ a n a n J

hence

Z* v- ( r ) - 4 Vs( r ) f A k--lu(y)jdc,

= Z Z [ AJy V , ( r ) ~ A k- i u ( y ) - A t u ( y ) f A k- i V,(r))dor

k =0 j = 0 s R \ a r a r J

According to Lemma 1 we obtain

Z* Z < J [Ai Vs ( r ) f A k- J u ( y ) - A J u ( y ) f A k- J V,(r))dar

k = 0 j ~ 0 SR \ а Г a r J

= z ‘ Z « f ( c * Z ( - t f f i ) * .( v . - , < r ) ± # - > « № , -

к = 0 j = 0 \ » = 0 V / S R а Г

- c 2(k~j) £ ( - I f f W - I 7 “ K - i ( r) Aj u (y)d°y)= A l + A

,

i =0 v ' s R a r /

where

A i

= Z Z Z

J

j u(y)doy,

k = 0 j = 0 i = 0 S R

A 2

= " z Z

- 1>*(*-Ла,. J ( e r r 1* , - , - .

k =0j = 0 i = 0

V 1 ) sR

(cr) Aj и (у) do у.

266 J. G ô r o w s k i , K. W a c h n i c k a

Let A t — A \ + A X+ A ^ , where

A \ denotes the sum of these ingredients of the sum A x for which s — i > 0, A® denotes the sum of these ingredients of the sum A x for which s — i = 0, A \ denotes the sum of these ingredients of the sum A x for which s — i < 0 . Now we shall prove that A x , A x, A f -> 0 as R -> 0. We need the following formulas ([3], p. 79):

K,(z) * A„(z, * =

2 ~»0 + Z9 2-0 +

q > 0

For s — i > 0 we get

f ( c r r i K s. i (cr)~-Ak- j u(y)da)

sR ar

thus A x -> 0 as R -> 0.

If s — i = 0, then

< M 1(cR)s- i K s_i{cR)Rn- 1

^ M 2 R n 1 -> 0 as R -► 0,

«я "r

Hence /1° -> 0 as R - * 0 . For s — i < 0 we have

< < М 3(1+|1пД|)Д*

Sr a r

^ M 1( c Rf ~ i K i^ s(cR)Rn- 1

^ M4 R2p_2,'“ 1.

Since / <r p — 1, we have 2p — 2/ — 1 > 0. Hence y4f 0 as R ->0.

The numbers M,, / = 1, 2, 3, 4 denote the positive constants.

Let Л2 = A 2 + A 2 + A 2 , where A 2 , A 2, A 2 denote the sum of these ingredients of the sum A 2 for which s — i — 1 > 0, s — i — 1 = 0 , s — i — 1 < 0, respectively.

Similarly, we can verify that A 2 , A 2 - ^ 0 as R -> 0.

Let A 2 = /421 + ^ 22^ where A 21, A 22 denote sums of these ingredients in the sum A 2 for which i Ф p — 1, i = p — 1, respectively.

If л — / — 1 < 0 and / Ф p ~ U then

I J (cr)s ' K s- i- l (cr)AJu(y)d<ry\ ^ Mf( cR) s ' K _s + I + 1 (cR)Rn 1 S

^ M $ R 2p~ 2j- 2, where M *, M f are positive constants.

Since i = 0 ,1 ,..., k —j, j = 0, 1 ,..., к, к — 0 ,1 ,..., p — 1, the condition

i Ф p — 1 is equivalent to the condition j < p ~ 1. In this case 2p — 2i — 2 > 0 and A

-> 0 as R -> 0.

Let s — / — 1 < 0 and i = p — 1. This case is true if and only if к = p — 1 and

j

= 0. Hence

Л

= a ^ 1c2<’- 1( - i r 4 - 1(cR)1- ',,2K_„;2(cR) J u(y)do,.

SR We remark that

(cR)n/2 K n/2(cR) 2"/2_ 1 Г (n/2) as R -> 0.

Moreover,

---L-rj j u{y)doy ->w(x) as

ы п К s R

where cu„ denotes the surface area of the unit sphere in Æ". Hence

^ 22- >с2р_и( - 1)р" 1( р - 1)!2р+,,/2" 2Г(п/2)£и||и(х) as R-+Q.

We find Г(п/2)а>„ = 2nn/2, thus

Л72 ->с2р" и( - 1 ) р“ 1(р - 1 ) ! 2 р+п/2" 1я ,,'2м(х) as К ->0.

Finally

Л1+ Л 2 ^ с 2р“ " ( - 1 ) р“ 1(р -1 )!2 р + ,,/2“ 17г,,/2м(х) as R -> 0, which ends the proof of Theorem 1.

Similarly we can prove

T

2. Let D be a bounded domain whose boundary we denote by S. We assume that S consists o f a finite number of piecewise-smooth hypersurfaces. I f the function u e C 2p(D) is continuous with the derivatives up to the order 2p—l in

D

S and (A + c2f и (x) = 0 in D, then

Ï L K U d Ju ( y ) ^ A l- l( ( c r Y N - A c r ) ) - 4 ( ( c r r N ^ ( c r ) ) - !l- A k- l u (y ))lt';>

k = 0 j = 0 S \ а П а П J

(0 for x e R n\(D u S ),

~ \ { - l f ( p - l ) \ c 2p- n2p+nl2nnl2- l u{x) for

where fip k — l ^ c 2p~ 2k~ 2.

3. We consider the following boundary value problem (A — c2f и (x) = 0 for x e D ,

(4) AJu(x) = fj(x) for x e S , j = 0, 1 , p - 1 ,

where f are given functions defined on S.

268 J. G ô r o w s k i , K. W a c h n i c k a

Let G(x, y) be Green function for problem (4) (we assume that it exists).

Under some assumptions on the function/) we can prove that the function u{x) = A ~ 1 £ Z <XkSfj(y)-j-tf-J'G(x,y)d(Ty

k =

0

=0

is a solution of problem (4), where

A = — 2p+nl2~ 1 (p—l)lc 2p~n.

We can obtain a similar formula for the equation (A + c2fu ( x ) = 0.

Moreover, using the Vekua results [1], [2], we can obtain a solution of the following problem

AP0( A + c f f l (A+C

f 2 - - - W + c ifku(x) = 0 for x e D ,

к

Aj u(x) = hj(x) for x e S ,

= 0, 1 ,

1.

i = 0

The analogous result can be obtained for the problem with boundary condition of the type

-^-AJu(x) = hj{x) for x e S ,

= 0, 1 , £ p , - l .

dn j=o

Hence, a solution of these problems can be reduced to the construction of respective Green’s functions.

References

[1] I. N. V ek ua, Metaharmonic functions, Trudy Tbiliss. Mat. Inst. 12 (1943), 105-174 (in Russian).

[2] —, New methods fo r solving elliptic equationn, Amsterdam 1967.

[3] G. N. W a ts o n , A treatise on the theory o f Bessel function, Cambridge 1962.