V.A. Steklov and the Problem of Sharp (Exact) Constants in Inequalities of Mathematical Physics

Nazarov, Alexander I.; Kuznetsov, Nikolay; Poborchi, Sergei V

doi:10.48550/arxiv.1307.8025

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2015

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“…The studies of Steklov-type eigenvalue problems are related to several important problems in differential geometry, see for examples, [13,14,35,17,18]. They are also closely connected with some classical geometric inequalities and Sobolev trace inequalities, see [12,14,31,32,36,37,43,44]. It is also wellknown that the Dirichlet to Neumann map is an essential tool for studies of many inverse problems.…”

Section: Introductionmentioning

confidence: 99%

Nodal sets of Steklov eigenfunctions

Bellová

Lin

2015

Calc. Var.

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We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in R n -the eigenfunctions of the Dirichlet-toNeumann map. Under the assumption that the domain Ω is C 2 , we prove a doubling property for the eigenfunction u. We estimate the Hausdorff H n−2 -measure of the nodal set of u| ∂Ω in terms of the eigenvalue λ as λ grows to infinity. In case that the domain Ω is analytic, we prove a polynomial bound O(λ 6 ). Our arguments, which make heavy use of Almgren's frequency functions, are built on the previous works [Garofalo and Lin, CPAM 40 (1987), no. 3; Lin, CPAM 42 (1989), no. 6].

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