Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

TL;DR

This paper develops a method to derive precise spectral asymptotics for Steklov problems with corners, focusing on the two-dimensional sloshing problem, confirming a long-standing conjecture and extending results to related problems.

Contribution

It introduces a new approach to obtain sharp asymptotics for Steklov eigenvalues on domains with corners, confirming a conjecture and broadening understanding of spectral behavior.

Findings

Two-term asymptotic formula for sloshing eigenvalues

Confirmation of Fox and Kuttler's 1983 conjecture

Eigenvalue asymptotics for related Steklov problems

Abstract

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov-Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.