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Hearing the shape of membranes: Further results. (English) Zbl 0717.35093
The spectral function \(\theta (t)=\sum \exp (-t\lambda_ m)\), \(t>0\), where \(\{\lambda_ m\}\) are the eigenvalues of the Laplacian in \({\mathbb{R}}^ n\), \(n=2\) or 3, is studied for a variety of domains, especially relative to the boundary condition (see impedance tomography) \(\partial u/\partial n+\gamma_ ju=0\) on \(\Gamma_ j\) or \(S_ j\), \(j=1,...,J\). Here \(\Gamma_ j\subset {\mathbb{R}}^ 2\) and \(S_ j\subset {\mathbb{R}}^ 3\) are parts of the surface of a ball. \(\theta\) is represented by means of Green’s function of the heat equation relative to these boundary conditions. There are generalizations to other domains.
Reviewer: G.Anger
MSC:
35R30 Inverse problems for PDEs
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
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