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English Pages 196 Year 1997
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|