Higher dimensional inverse problem of the wave equation for a general multi-connected bounded domain with a finite number of smooth mixed boundary conditions. (English) Zbl 1033.35152
Summary: The spectral function \(\widehat\mu(t)= \sum^\infty_{J=1} \exp(- it\mu^{1/2}_J)\) where \(\{\mu_J\}^\infty_{J=1}\) are the eigenvalues of the negative Laplacian \(-\Delta_3= -\sum^3_{v=1} (\partial/\partial x^v)^2\) in the \((x^1,x^2,x^3)\)-space, is studied for small \(| t|\) for a variety of bounded domains, where \(-\infty< t< \infty\) and \(i= \sqrt{-1}\). The dependences of \(\widehat\mu(t)\) on the connectivity of bounded domains and the boundary conditions are analysed. Particular attention is given to a general multi-connected bounded domain \(\Omega\) in \(\mathbb{R}^3\) together with a finite number of smooth Dirichlet, Neumann and Robin boundary conditions on the smooth boundaries \(\partial\Omega_J\) \((J= 1,\dots, m)\) of the domain \(\Omega\). Some geometrical properties of \(\Omega\) (e.g., the volume, the surface area, the mean curvature and the Gaussian curvature of \(\Omega\)) are determined from the asymptotic expansions of \(\widehat\mu(t)\) for small \(| t|\).
MSC:
| 35R30 | Inverse problems for PDEs |
| 35L05 | Wave equation |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
Keywords:
inverse spectral problem; wave equation; multi-connected domain; spectral function; Green’s function; fractal boundary; eigenvalues of the negative LaplacianReferences:
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