Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons
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.
Keywords
Cite
@article{arxiv.1908.06455,
title = {Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons},
author = {Michael Levitin and Leonid Parnovski and Iosif Polterovich and David A. Sher},
journal= {arXiv preprint arXiv:1908.06455},
year = {2022}
}
Comments
Final accepted version. 118 pages; 24 figures; continues the programme initiated in arXiv:1709.01891