Levitin, Michael;
Parnovski, Leonid;
Polterovich, Iosif;
Sher, David A;
(2022)
Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons.
Proceedings of the London Mathematical Society
10.1112/plms.12461.
(In press).
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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.
| Type: | Article |
|---|---|
| Title: | Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1112/plms.12461 |
| Publisher version: | http://dx.doi.org/10.1112/plms.12461 |
| Language: | English |
| Additional information: | This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third-party material in this article are included in the Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ |
| Keywords: | Science & Technology, Physical Sciences, Mathematics, SHORT SURFACE-WAVES, SPECTRAL GEOMETRY, QUANTUM GRAPHS, OPERATORS, DOMAINS, DEPENDENCE, FREQUENCY, EQUATION |
| UCL classification: | UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics UCL > Provost and Vice Provost Offices > UCL BEAMS UCL |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10151388 |
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