Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons
arXiv:1908.06455 · doi:10.1112/plms.12461
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.
Final accepted version. 118 pages; 24 figures; continues the programme initiated in arXiv:1709.01891
References in corpus (8)
- Index theorems for quantum graphs
- Optimization of Steklov-Neumann eigenvalues
- The heat kernel on curvilinear polygonal domains in surfaces
- Effective operators for Robin eigenvalues in domains with corners
- Inverse Steklov spectral problem for curvilinear polygons
- Heat trace asymptotics for quantum graphs
- Spectral geometry of the Steklov problem
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