Some results on higher eigenvalue optimization
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Calculus of Variations
Some results on higher eigenvalue optimization Ailana Fraser1 · Richard Schoen2 Received: 9 October 2019 / Accepted: 30 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) k-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for k ≥ 3. For k = 1 the classical result of Weinstock (J Ration Mech Anal 3:745–753, 1954) shows that σ1 is maximized by the standard metric on the round disk. For k = 2 it was shown by Girouard and Polterovich (Funct Anal Appl 44(2):106–117, 2010) that σ2 is not maximized for a smooth metric. We also prove a local rigidity result for the critical catenoid and the critical Möbius band as free boundary minimal surfaces in a ball under C 2 deformations. We next show that the first k Steklov eigenvalues are continuous under certain degenerations of Riemannian manifolds in any dimension. Finally we show that for k ≥ 2 the supremum of the k-th Steklov eigenvalue on the annulus over all metrics is strictly larger that that over S 1 -invariant metrics. We prove this same result for metrics on the Möbius band. Mathematics Subject Classification 35P15 · 53A10
1 Introduction In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. Recall that for a compact Riemannian manifold
Communicated by P. Topping. A. Fraser was partially supported by the Natural Sciences and Engineering Research Council of Canada and R. Schoen was partially supported by NSF Grant DMS-1710565. Part of this work was done while the authors were visiting the Institute for Advanced Study, with funding from NSF Grant DMS-1638352 and the James D. Wolfensohn Fund, and the authors gratefully acknowledge the support of the IAS.
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Ailana Fraser [email protected] Richard Schoen [email protected]
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Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
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Department of Mathematics, University of California, Irvine, CA 92617, USA 0123456789().: V,-vol
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with non-empty boundary we have the Steklov spectrum which consists of the eigenvalues of the Dirichlet to Neumann map. We denote these eigenvalues σ0 = 0 < σ1 ≤ σ2 . . . and they form an infinite discrete sequence tending to infinity. A Steklov eigenfunction u with eigenvalue σ is then a non-zero solution of u = 0 in M with ∂u ∂ν = σ u on ∂ M where ν denotes the outward unit normal to ∂ M. A classical result of Hersch et al. [16] from 1975 gives the upper bound σk · L(∂ D) ≤ 2πk for all metrics on the disk D and for all k ≥ 1. In 2010 it was shown by Girouard and Polterovich [14] that this bound is sharp for all k but is not attained by a smooth metric on the disk for k = 2. The bound and the result that it is attained by the standard round disk f
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