Direct and inverse problems for a Schrödinger-Steklov eigenproblem on different domains and spectral geometry for the first normalized Steklov eigenvalue on domains with one hole

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2018-01-01
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Rodriguez-Quinones, Leoncio
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Hien Nguyen
Paul E. Sacks
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Mathematics
Abstract

We present some results related with the asymptotic expansion of the eigenvalues for the Schr\"{o}dinger-Steklov eigenvalue problem. We find explicit expressions for this asymptotic behavior in the cases when the domain is the unit disk in $\mathbb{R}^{2}$ and the unit ball in $\mathbb{R}^{3}$. We also show a uniqueness result for the potential function $q$ based on the knowledge of the Steklov spectrum. Several numerical examples are included to illustrate different strategies in the recovery of the potential function $q$ from the knowledge of some of the eigenvalues. Finally, we describe a shape derivative approach to provide a candidate for an optimal domain in the class of bounded planar domains with one hole for the first nontrivial Steklov eigenvalue. This approach is an adaptation of the work on the extremal eigenvalue problem for the Wentzell-Laplace operator developed by Dambrine, Kateb and Lamboley.

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Wed Aug 01 00:00:00 UTC 2018