Spherical Means for PDEs [Reprint 2016 ed.] 9783110926026, 9783110460568

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SPHERICAL M E A N S

FOR PDEs K.K. Sabelfeld

and I.A.

S//VSP/// Utrecht, The Netherlands, 1997

Shalimova

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

© V S P BV 1997 First published in 1997 ISBN 90-6764-211-8

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands by Ridderprint bv,

Ridderkerk.

Contents

1. Introduction

1

2. Scalar second order PDEs

5

2.1. Spherical mean value relations for the Laplace equation and integral formulation of the Dirichlet problem 2.1.1. Direct spherical mean value relation 2.1.2. Converse mean value theorem

5 6 13

2.1.3. Integral equation equivalent to the Dirichlet problem 15 2.1.4. Poisson-Jensen formula 2.2. The diffusion and Helmholtz equations

17 19

2.2.1. Diffusion equation

20

2.2.2. Helmholtz equation

21

2.3. Generalized second order elliptic equations

23

2.4. Parabolic equations

26

2.4.1. Heat equation

26

2.4.2. Parabolic equations with variable coefficients . . .

32

2.4.3. Expansion of the parabolic means

35

3. High-order elliptic equations

39

3.1. Balayage operator

39

3.2. The biharmonic equation

41

3.2.1. Direct spherical mean value relation

42

3.2.2. The generalized Poisson formula

44

3.2.3. Rigid fixing of the boundary

49

3.2.4. Nonhomogeneous biharmonic equation

52

Contents

IV

3.3. Fourth order equation governing the bending of a plate on an elastic base surface

54

3.4. Metaharmonic equations

59

3.4.1. Polyharmonic equation

59

3.4.2. General case

62

4. Triangular systems of elliptic equations

68

4.1. A one-component diffusion system

68

4.2. A two-component diffusion system

70

4.3. A coupled biharmonic-harmonic equation

73

5. Systems of elasticity theory

75

5.1. The Lamé equation

75

5.1.1. Direct spherical mean value theorem

75

5.1.2. Converse spherical mean value theorem

81

5.2. Pseudo-vibration elastic equation

84

5.3. Thermo-elastic equation

92

6. T h e generalized Poisson formula for t h e Lamé equation 6.1. Plane elasticity

94 94

6.2. Generalized spatial Poisson formula for the Lamé equation 105 6.3. An alternative derivation of the Poisson formula 7. Spherical means for the stress and strain tensors

122 127

7.1. Spherical means for the displacement components through the displacement vector 127 7.2. Mean value relation for the stress and strain tensors through the displacement vector 131 7.2.1. Mean value relation for the strain components . . 131 7.2.2. Mean value relation for the stress components

. . 137

7.3. Mean value relations for the stress components in terms of the surface tractions 138

V

Contents

8. Applications t o the R a n d o m Walk on Spheres m e t h o d

150

8.1. Spherical mean as a mathematical expectation

150

8.2. Iterations of the spherical mean operator

151

8.3. The Random Walk on Spheres algorithm

152

8.3.1. The Random Walk on Spheres process for the Dirichlet problem 8.3.2. Inhomogeneous case

152 164

8.4. Biharmonic equation

166

8.5. Random Walk on Spheres method for the Lamé equation

168

8.5.1. Naive generalization

168

8.5.2. A modification of the algorithm

170

8.5.3. Non-isotropic Random Walk on Spheres

173

8.5.4. Branching process 175 8.5.5. Analytical continuation with respect to the spectral parameter 178 8.6. Alternative Schwarz procedure Bibliography

181 185

vii

List of Symbols Euclidean space of dimension n

]R n G

a domain in E "

G

the closure of G

T =

the boundary of G

dG

distance from a point x to the boundary

d(x)

r a ball of radius r centered at the point

B(x,r)

x S(x,r)

=

dB{x,r)

a sphere of radius r centered at the point x a disk of radius r centered at x

K(x,r) 5(x,r) =

dK(x,r)

a circle of radius r centered at x

fi o. = £lz£L S' R

the unit sphere 5 ( 0 , 1 )

dQ.

the surface element of Q

dSy = dS(y)

coordinates of a unit directional vector s

= dS = do

y the Euler gamma-function

T(m)

the area of the surface of the unit sphere

um = 2 W 2 / r ( m / 2 )

Nru(x) = iv s 0 =

the surface element of 5(0, r ) at a point

7Vr(u)

in ]R m the spherical mean of the function u(x) the volume average of 6 over a ball binomial coefficients

Au, = Vijj

the Laplace operator of a function V{

A* — fj,A + (A + M) g r a d div

the Lamé operator

u

the displacement vector, a solution to the Lamé equation A * u = 0

Vili — 2 (^iJ

the components of the strain tensor

r!;- = 2fi£ij + A

x

(2.1)

eG y e

(2.2)

r

We seek a regular solution to (2.1),(2.2), i.e., u £

C2{G)^\C{G).

6

2. Scalar second order PDEs

2.1.1.

Direct spherical mean value relation

It is well known that every regular solution to (2.1) satisfies the spherical mean value relation: u{x) = Nru(x) := — [ u(x + rs) dtt(s) Wm J fi

(2.3)

for each x 6 G and for all spheres S(x,r) contained in G := G |J T. The same is true for the volume mean value relation (it can be obtained directly from (2.3) by integrating): f J u(y) dy.

771

u(x) = —

(2.4)

B{x,r)

The mean value relation (2.3) can be derived by different methods. For small r, it is possible to use the method based on the power expansion of the integrand. We present this method here, and we will use it later to derive the mean value relations for different equations. The following statement is very useful, in particular, to get power expansions of the spherical means. We denote by D the differential operator D = ( — i af-) V * / Da = Dy1 • • • D%>, where a is the multiindex: a = (c*i,...

,am),a!

= a^!. ..am\,Dk

=

l

33 (*> *) G i(x, < ) , . . . , bn(x, t)). The function Z(x, t; x't') solves the equation dZ dZ - ^ 7 - H & x > Z ( x , t , x ' , t ' ) - b i ( x , t ) — { x , t - , x l , t ' ) = 0,

t'

0,

/ > 0,

k + 21 < 2p .

Theorem 2.10. Assume that u(x, t) G C2p+2^+l(G), p > 0. Then for all inner points (x, t) £ G and for all sufficiently small values r (such that 0 < r < d(x, t)) the following expansion is true (Mru)(x, where the differential Li =

(~iy

{f

p t) = Y^ r2iLiu(x, ¿=o operators

m nn ++ m

t) + Qp(r)u(x,

are defined by

\ (»+»0/2+1 /d_ _

\n + m + 2i)

(2.66)

t),

nn ++ mm

vV-

n + m + 2i

J

i (2.67)

2-4• Parabolic for

i >1,

I f u(x,

L0

= 1, and

t) £ C°°(G)

= 0 ( r 2 " + 1 ) as r

Qp(r)u(x,t)

37

equations 0.

satisfies the condition

lim Qp(r)u(x, p—> oo

t)

= 0,

(2.68)

then we can pass to the limit as p —> oo. Thus for the solutions to the equation ^ = Lu(x,t) satisfying (2.68), we have the following mean value relation u(x,t) = (Mru)(x,t) (2.69) for all (x,t) G G and all r, 0 < r < r(x,t). It is not difficult to show that this property is true for each solution of ^ = Lu from the class C2-1. Conversely, if a function u G C 4 ' 2 satisfies (2.69), then we get from the expansion that 0 = r2Lu(x,t)

+

0(rz),

or 0 = Lu(x, t)+0(r). Then we get by r —• 0: Lu(x, t) = 0. This implies the weak mean value theorem. A strong variant of this statement can be proved using the maximum principle according to the scheme given for the Laplace equation. Note that different expansions of the type (2.66) can be derived. For instance, choosing i K(y,r)

=

a| 1 /2 7r n/ 2 2 n+m r(m/2)r n+m2+2ra

x T (" ,2 + m )/ 2 - 1 [2(n + m2 + 2m)r ln(r 2 /r) -

yTa~ly]m^-1

x [(m + 2)(2(n + m 2 + 2m)r ln(r 2 /r) - j/ T a _1 y) + yTa~1y] (2.70) and Br

= { ( y , r ) : yTa~1y

< 2(n + m2 + 2 m ) r ln(r2/r)}

we come to an expansion of the type (2.66) with

38

2. Scalar second order PDEs

Li = ^ [ 9 ( n M \ i n + m W { § - t - 9 ( n , r n ) L J where . gin,m)

.

=

*

-

,

(2.71)

n + mr + 2 m -— — .

Note that both expansions with (2.66) and (2.71) result, when m —> +0, in the expansions with 72) This is the expansion of the average of the function u(x,t) surfaces (spheroids) { ( y , r ) : Z(x, t, y, r ) = Tr^M"1/2»--"}

.

over the

Chapter 3

High-order elliptic equations

3.1.

Balayage operator

In this section we give a general scheme of construction of the spherical mean value relations for high-order elliptic equations (see [37] and [38]). Let X be a locally compact Hausdorf space, and let CQ(X) be a space of continuous functions / : X —* H with a compact support. We denote by R(X) the space of Radon measures FI : CQ{X) —• JR. Let G C 1R" be a bounded domain with the boundary dG = T. By Ck(G) we denote the space of continuous on G functions u(x) such that the derivatives Dau exist in G for all |a| < k and admit continuous continuation to G. Let Cq°(G) be a space of infinitely differentiate functions with a compact support in G. Introduce a vector measure /i = (/i tt )H< t e

(g) R{ R " ) , |a|