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Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

Michael Levitin, Leonid Parnovski, Iosif Polterovich, David A. Sher

Abstract

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.

Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

Abstract

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.

Paper Structure

This paper contains 41 sections, 36 theorems, 248 equations, 6 figures.

Table of Contents

  1. Introduction and main results
  2. The sloshing problem
  3. Asymptotics of the sloshing eigenvalues
  4. Eigenvalue asymptotics for a mixed Steklov-Dirichlet problem
  5. Outline of the approach
  6. An application to higher order Sturm-Liouville problems
  7. Towards sharp asymptotics for Steklov eigenvalues on polygons
  8. Plan of the paper
  9. Construction of quasimodes for triangular domains
  10. The sloping beach problem
  11. Quasimode analysis
  12. Exponentially accurate quasimodes for angles of the form $\frac{\pi}{2q}$
  13. Hanson-Lewy solutions for angles $\pi/2q$
  14. Exponentially accurate quasimodes for angles of the form $\pi/2q$
  15. Completeness of quasimodes via higher order Sturm-Liouville equations
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $\Omega$ be a simply connected bounded Lipschitz planar domain with the sloshing surface $S=(A,B)$ of length $L$ and walls $\mathcal{W}$ which are $C^1$-regular near the corner points $A$ and $B$. Let $\alpha$ and $\beta$ be the interior angles between $\mathcal{W}$ and $S$ at the points $A$ and If, moreover, the walls $\mathcal{W}$ are straight near the corners, then

Figures (6)

  • Figure 1: Geometry of the sloshing problem
  • Figure 2: Geometry of the sloshing problem with walls straight near the corners
  • Figure 3: A polygon with auxiliary curves $\mathcal{L}$
  • Figure 4: Construction of auxiliary triangles in the case $\alpha=\pi/2$ (left) and $\alpha<\pi/2$ (right)
  • Figure 5: Triangle from Example \ref{['sec:ex1']}
  • ...and 1 more figures

Theorems & Definitions (82)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 72 more