An inverse problem for the three-dimensional multi-connected vibrating membrane with Robin boundary conditions

Author:
E. M. E. Zayed

Journal:
Quart. Appl. Math. **61** (2003), 233-249

MSC:
Primary 35P20; Secondary 35J40, 35R30

DOI:
https://doi.org/10.1090/qam/1976367

MathSciNet review:
MR1976367

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Abstract: This paper deals with the very interesting problem concerning the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in . The trace of the heat semigroup , where are the eigenvalues of the negative Laplacian in the -space, is studied for a general multiply-connected bounded domain in surrounding by simply connected bounded domains with smooth bounding surfaces , where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components of the bounding surfaces is considered, such that , where . Some applications of for an ideal gas enclosed in the multiply-connected bounded container with Robin boundary conditions are given. We show that the asymptotic expansion of for short-time plays an important role in investigating the influence of the finite container on the thermodynamic quantities of an ideal gas.

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DOI:
https://doi.org/10.1090/qam/1976367

Article copyright:
© Copyright 2003
American Mathematical Society