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Higher dimensional inverse problem for a multi-connected bounded domain with piecewise smooth Robin boundary conditions and its physical applications
Faculty
Science
Year:
2003
Type of Publication:
Article
Pages:
365-379
Authors:
ZAYED, EME
DOI:
10.1016/S0096-3003(02)00413-7
Journal:
APPLIED MATHEMATICS AND COMPUTATION ELSEVIER SCIENCE INC
Volume:
144
Research Area:
Mathematics
ISSN
ISI:000184353800013
Keywords :
inverse problem, heat kernel, eigenvalues, on hearing the shape of drum, classical ideal gas
Abstract:
The asymptotic expansions of the trace of the heat kernel Theta(t) = Sigma(j=1)(infinity) exp(-tlambda(j)) for short-time t, have been derived for a variety of bounded domains, where are the eigenvalues of the negative Laplace operator -del(2) = -Sigma(i=1)(3)(partial derivative/partial derivativex(i))(2) in the (x(1), x(2), x(3))- space. The dependence of Theta(t) on the connectivity of bounded domains and the boundary conditions is analyzed. Particular attention is given for an arbitrary multiply-connected bounded domain Omega in R-3 together with piecewise smooth Robin boundary conditions, where the coefficients in these conditions are assumed to be piecewise smooth positive functions. Some applications of an ideal gas enclosed in the multiply-connected bounded domain Omega are given. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it. (C) 2002 Elsevier Inc. All rights reserved.
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