Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space
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We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: We prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk.
KeywordsSteklov problem Euclidean space Prescribed boundary manifolds Hypersurfaces of revolution
Mathematics Subject Classification35P15 58C40
We thank one of the anonymous referees for suggesting an elegant proof to Proposition 1.10. We thank another referee for carefully reading the paper and suggesting several improvements, in particular regarding Theorem 1.11. While a postdoctoral student at the Université de Neuchâtel, KG was supported by the Swiss National Science Foundation Grant No. 200021_163228 entitled Geometric Spectral Theory. KG also acknowledges support from the Max Planck Institute for Mathematics, Bonn. AG acknowledges support from the NSERC Discovery Grants Program.
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