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 inR 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 inR ³ together with its smooth inner boundary surfaceS ₁ and its smooth outer boundary surfaceS ₂. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth componentsS ^* _i(i=1, …,m) ofS ₁ and on the piecewise smooth componentsS ^* _i(i=m+1, …,n) ofS 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⋎.