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In this paper, we study the bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the boundary and the diameter respectively. We also prove similar estimates for higher order Steklov eigenvalues.
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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.
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.
We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,... The equality is attained in the limit by a sequence of simply-connected domains degenerating to the disjoint union of n identical disks. We give a new proof of this inequality for n=2 and show that it is strict in this case. Related results are also obtained for the product of two consecutive Steklov eigenvalues.
We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity of the first nonzero Steklov eigenvalues on trees: for finite trees $g_1$ and $g_2,$ the first nonzero Steklov eigenvalue of $g_1$ is greater than or equal to that of $g_2$, provided that $g_1$ is a subgraph of $g_2.$
We prove two explicit bounds for the multiplicities of Steklov eigenvalues $sigma_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Riemannian surface with boundary, the multiplicities of Steklov eigenvalues $sigma_k$ are uniformly bounded in $k$.
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