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Spectral geometry of the Steklov problem. (English) Zbl 1375.49056

Henrot, Antoine (ed.), Shape optimization and spectral theory. Berlin: De Gruyter (ISBN 978-3-11-055085-6/hbk; 978-3-11-055088-7/ebook). 120-148 (2017).
Let \(\Omega\) be a compact Riemannian manifold with smooth boundary. The Steklov problem is \(\Delta u=0\) and \(\partial_nu=\sigma u\) on \(\partial\Omega\) where \(\Delta\) is the Laplacian and \(\partial_n\) is the outward unit normal. The spectrum is discrete assuming the trace operator \(H^1(\Omega)\rightarrow L^2(\partial\Omega)\) is compact and the Steklov eigenvalues form a discrete sequence \(0=\sigma_1\leq\sigma_2\leq\dots\) where \(\sigma_n\rightarrow\infty\) assuming, for example, the boundary of \(\Omega\) is Lipschitz. The Steklov eigenvalues can also be interpreted as the eigenvalues of the Dirichlet to Neumann operator. In Section 5.1, the authors motivate the problem at hand, review the history of the problem, and provide some computational examples. Section 5.2 is a survey of results on asymptotics of the spectrum. Section 5.3 deals with polygonal regions and Section 5.4 presents some geometric inequalities. Section 5.5 treats Steklov isospectral and spectral rigidity and Section 5.6 discusses nodal geometry and multiplicity bounds.
For the entire collection see [Zbl 1369.49004].

MSC:

49Q10 Optimization of shapes other than minimal surfaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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