Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces

TL;DR

This paper explores the use of hidden symmetries to analyze Steklov and mixed Steklov problems on surfaces, deriving new inequalities, examining maximization properties of eigenvalues, and providing asymptotic results under boundary conditions.

Contribution

It introduces novel symmetry-based techniques to study Steklov eigenvalues, establishes a sharp isoperimetric inequality, and investigates eigenvalue maximization in symmetric domains.

Findings

Sharp isoperimetric inequality for mixed Steklov eigenvalues

Disk maximizes eigenvalues for large k in symmetric domains

Asymptotic formulas for mixed Steklov problems on surfaces

Abstract

We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces we study do not necessarily have inherent symmetry. In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric inequality for the mixed Steklov eigenvalues considering the interplay between the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues. In 1980, Bandle showed that the unit disk maximizes the $k$th nonzero normalized Steklov eigenvalue on simply connected domains with rotational symmetry of order $p$ when $k\le p-1$. We discuss whether the disk remains the maximizer in the class of simply connected rotationally symmetric domains when $k\geq p$. In particular, we show that for $k$ large enough, the upper bound converges to the Hersch-Payne-Schiffer upper bound. We give full asymptotics for mixed Steklov problems on arbitrary surfaces, assuming some conditions at the meeting points of the Steklov boundary with the Dirichlet or Neumann boundary.