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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Compact manifolds with fixed boundary and large Steklov eigenvalues
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by Bruno Colbois, Ahmad El Soufi and Alexandre Girouard PDF
Proc. Amer. Math. Soc. 147 (2019), 3813-3827 Request permission

Abstract:

Let $(M,g)$ be a compact Riemannian manifold with boundary. Let $b>0$ be the number of connected components of its boundary. For manifolds of dimension $\geq 3$, we prove that for $j=b+1$ it is possible to obtain an arbitrarily large Steklov eigenvalue $\sigma _j(M,e^\delta g)$ using a conformal perturbation $\delta \in C^\infty (M)$ which is supported in a thin neighbourhood of the boundary, with $\delta =0$ on the boundary. For $j\leq b$, it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of $M$. In fact, when working in a fixed conformal class and for $\delta =0$ on the boundary, it is known that the volume of $(M,e^\delta g)$ has to tend to infinity in order for some $\sigma _j$ to become arbitrarily large. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.
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Additional Information
  • Bruno Colbois
  • Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland
  • MR Author ID: 50460
  • Email: bruno.colbois@unine.ch
  • Ahmad El Soufi
  • Affiliation: Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 6083, Université François Rabelais de Tours, Parc de Grandmont, 37200 Tours, France
  • Alexandre Girouard
  • Affiliation: Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Québec, Québec, G1V 0A6, Canada
  • MR Author ID: 832728
  • Email: alexandre.girouard@mat.ulaval.ca
  • Received by editor(s): February 2, 2017
  • Received by editor(s) in revised form: October 14, 2018
  • Published electronically: May 17, 2019
  • Additional Notes: During the first week of 2017, the first and third authors were supposed to travel to Tours and work with Ahmad El Soufi to complete this paper. We learned just a few days before our visit of his untimely death. Ahmad was a colleague and a friend. He will be dearly missed.
  • Communicated by: Michael Wolf
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 3813-3827
  • MSC (2010): Primary 35P15, 53C21, 58J50; Secondary 53C23, 53C20
  • DOI: https://doi.org/10.1090/proc/14426
  • MathSciNet review: 3993774