# An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane with piecewise smooth boundary conditions

## Abstract

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian in*R* ^{3}. The asymptotic expansion of the trace of the wave operator\(\widehat\mu (t) = \sum\limits_{\upsilon = 1}^\infty {\exp \left( { - it\mu _\upsilon ^{1/2} } \right)} \) for small ⋎*t*⋎ and\(i = \sqrt { - 1} \), where\(\{ \mu _\nu \} _{\nu = 1}^\infty \) are the eigenvalues of the negative Laplacian\( - \nabla ^2 = - \sum\limits_{k = 1}^3 {\left( {\frac{\partial }{{\partial x^k }}} \right)} ^2 \) in the (*x* ^{1},*x* ^{2},*x* ^{3}), is studied for an annular vibrating membrane Ω in*R* ^{3} together with its smooth inner boundary surface*S* _{1} and its smooth outer boundary surface*S* _{2}. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components*S* ^{*} _{i}(*i*=1, …,*m*) of*S* _{1} and on the piecewise smooth components*S* ^{*} _{i}(*i=m*+1, …,*n*) of*S* _{2} such that\(S_1 = \bigcup\limits_{i = 1}^m {S_i^* } \) and\(S_2 = \bigcup\limits_{i = m + 1}^n {S_i^* } \) are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane ω from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function\(\widehat\mu (t)\) for small ⋎*t*⋎.

## AMS Mathematics Subject Classification

Primary 35Kxx 35Pxx## Key words and phrases

Inverse problem, wave equation, annular vibrating membrane elgenvalues plecewise smooth boundary conditions spectral function heat kernel## References

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