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The wave equation approach to an inverse problem for a general multiconnected domain in \(\mathbb{R}^2\) with mixed boundary conditions. (English) Zbl 1080.35555

Summary: The spectral function \(\widehat\mu(t)= \sum^\alpha_{j=1}\exp(-it \mu_J^{1/2})\) where \(\{\mu_J\}^\infty_{J=1}\) are the eigenvalues of the negative Laplacian \(-\Delta_2=-\sum^2_{\nu=1}(\frac{\partial}{\partial x^v})^2\) in \(\mathbb{R}^2\) is studied for small \(|t|\) for a variety of domains, where \(-\infty<t<\infty\) and \(i=\sqrt{-1}\). The dependences of \(\widehat\mu (t)\) an the connectivity of domains and the boundary conditions are analyzed. Particular attention is given to a general multi-connected bounded domain in \(\mathbb{R}^2\) together with Dirichlet, Neumann and Robin boundary conditions on the boundaries \(\partial\Omega_J (J=1,\dots,m)\) of the domain \(\Omega\). Some geometrical properties of \(\Omega\) (e.g., the area of \(\Omega\), the total lengths of the boundaries \(\partial\Omega_J\), the cruvatures of \(\partial \Omega_J\), the numbher of holes of \(\Omega\), etc.) are determined from the asymptotic expansions of \(\widehat\mu(t)\) for small \(|t|\).

MSC:

35R30 Inverse problems for PDEs
35L05 Wave equation
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