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.

45 pages, 6 figures; a minor correction in the proof of Lemma A.4

References in corpus (6)

• The Steklov spectrum of cuboids
• Continuity properties of Neumann-to-Dirichlet maps with respect to the $H$-convergence of the coefficient matrices
• Victor Borisovich Lidskii (1924-2008)
• Inverse Steklov spectral problem for curvilinear polygons
• Spectral asymptotics for Dirichlet to Neumann operator
• Spectral geometry of the Steklov problem

Cited by in corpus (3)

• Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons
• Optimization of Steklov-Neumann eigenvalues
• Spectral asymptotics for Dirichlet to Neumann operator