Boundary value problems for harmonic random fields - Springer Link

B o u n d a r y V a l u e P r o b l e m s for H a r m o n i c Random Fields Y. Rozanov and F. Sansb

1

Introduction

We consider Laplace equation

A~t = 0

(1.1)

and boundary value problems for harmonic functions ( E W(G) of a certain class W(G) in a bounded region G C_ /i~d, whose behaviour near the boundary F = OG is as much chaotic as we see in a case of generalized random fields ~# &aa'acterized by the stochastic equation

A{ = r1

(1.2)

with a stochastic source r] of white noise type in C. The choice of W_W_(G)is motivated by a few important reasons as follows. We Would like to have the corresponding sample ~ E W(C) of chaotic boundacy behaviour in the case of (1.2) with any random Schwartz distribution

O0

*~

which is meansquare continuous over Schwartz test functions cfl with respect to L2-norm:

EI(~,~)I 2 ~ clI~It~ , ~ ~ c~(c)

(1.3)

68

We would like also to get a variety of stochastic boundary conditions such that the corresponding boundary data, given arbitrarily, determine a unique solution ~ E W___(G) of equation (1.1)/(1.2). And apart from of that, we would like not to fail in case, that the region G considered has inside a lot of singularities (totally of zero Lebesgue measure in Rd). These motivations lead us to stochastic Sobolev space.~

w(a)

= w

(a) , p = 2

of random Schwartz distributions

which are meansquare continuous over Schwartz test functions ~2 with respect to the corresponding norm

flail_p= (/'Ic)(.A)I2(I + IAI2)-~'dA) I/2

1.4)

given by means of the corresponding Fourier transform 1 Our approach to stochastic boundary problems for harmonic functions E W~(G), p = 2, is based on the application of an appropriate te~t

function space

x((;)

= [co

which appears as a closure of all Schwartz test functions p E Co~°(G) with respect to norm (1.4). 1A m a t e r i a l on classical ( d e t e r m i n i s t i c ) Sobolev spaces can be f o u n d in nearly any b o o k on P u n c t i o n a l A n a l i s e a n d PDE-see, for example, 1 e 2.

69 It occurs that in this way we get

x(G)

= w;-~(c;)

, p = 2

~s a space of all Schwartz distributions

= (~, ~)

, ~ ~ C J ° ( R d) ,

with supports suppz c_ [(;:] in the closure [G] of G, having a finite Ilorm

l{~{{x × II~{{-~ =

I~(A)I ~

i + IAI ~ -~ dA

, p = 2

(1.5)

Dealing with any ( E W(G), by meansquare continuity we can determine