Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov—Neumann boundary value problem describing small vertical oscillations of an i...

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Bibliographic Details
Published inJournal d'analyse mathématique (Jerusalem) Vol. 146; no. 1; pp. 65 - 125
Main Authors Levitin, Michael, Parnovski, Leonid, Polterovich, Iosif, Sher, David A.
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.08.2022
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Analysis
Asymptotic properties
Boundary value problems
Corners
Dynamical Systems and Ergodic Theory
Eigenvalues
Functional Analysis
Ideal fluids
Mathematics
Mathematics and Statistics
Partial Differential Equations
Vertical oscillations
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Summary:In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov—Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-021-0188-x