Isoperimetric control of the Steklov spectrum
By B. Colbois, Ahmad El Soufi, A. Girouard. 2011
107 citations recorded by the source corpus.
Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of the genus. We also obtain a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its bounding hypersurface.
View Paper: https://www.semanticscholar.org/paper/272bcf4231a1645d8c86859f736b4318e7347459
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- Mateusz Kwaśnicki
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Citations in the corpus
- SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “An interesting question raised in [20] is whether one can replace the exponent 2/n in Corollary 4.”
- Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem (2011, methodology): “The theorem below is motivated by the work of [CEG2], and we obtain an improvement of inequalities (26) and (27) using Proposition 3.”
- Upper bounds for Steklov eigenvalues on surfaces (2012, background): “In particular, a number of recent papers are concerned with the Steklov spectrum on manifolds with boundary [9, 10, 14, 7].”
- Variational aspects of Laplace eigenvalues on Riemannian surfaces (2011, unknown): “ preprint of the paper has appeared, there has b een a number of developments on the subject. First, the results in our Example 1.3 (the Steklov eigenvalueproblem) have been independently obtained in [7]. Extremal problems for Steklov eigenvalues have been also studied by Fraser and Schoen in [14] w”
- The legacy of Vladimir Andreevich Steklov (2014, background): “If the volume of ∂D is fixed, the corresponding Steklov eigenvalues λn of a Euclidean domain D can be bounded in terms of n and the dimension [8].”