# Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

## TL;DR

This paper derives precise asymptotic formulas for Steklov eigenvalues and eigenfunctions in curvilinear polygons, revealing the influence of angles and side lengths, with implications for spectral geometry and related physical problems.

## Contribution

It provides explicit asymptotic formulas for Steklov eigenvalues and eigenfunctions on curvilinear polygons, highlighting the role of angles and introducing new analytical techniques.

## Key findings

- Eigenvalues asymptotics depend on side lengths and angles.
- Angles significantly influence boundary eigenfunction behavior.
- Methods include explicit quasimode construction and spectral analysis.

## Abstract

We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.