Some results on higher eigenvalue optimization

• Published in: Calculus of variations and partial differential equations 2020-10, Vol.59 (5), Article 151
• Main Authors: Fraser, Ailana, Schoen, Richard
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Eigenvalues; Free boundaries; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Minimal surfaces; Optimization; Riemann manifold; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-020-01802-9

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 σ 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 σ 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 ≥ 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.