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 \(k\ge 3\). For \(k=1\) the classical result of Weinstock (J Ration Mech Anal 3:745–753, 1954) shows that \(\sigma _1\) 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 \(\sigma _2\) 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 \(k\ge 2\) 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.
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Communicated by P. Topping.
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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.
