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Sloshing, Steklov and corners: asymptotics of Steklov eigenvalues for curvilinear polygons

Levitin, M., Parnovski, L., Polterovich, I. and Sher, D. A. (2022) Sloshing, Steklov and corners: asymptotics of Steklov eigenvalues for curvilinear polygons. Proceedings of the London Mathematical Society. ISSN 1460-244X

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To link to this item DOI: 10.1112/plms.12461

Abstract/Summary

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.

Item Type:Article
Refereed:Yes
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:104988
Publisher:Wiley

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