Nodal Sets of Steklov Eigenfunctions

Katarina Bellova and Fanghua Lin

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Submission date: 18. Feb. 2014
Pages: 35
published in: Calculus of variations and partial differential equations, 54 (2015) 2, p. 2239-2268 
DOI number (of the published article): 10.1007/s00526-015-0864-8
Bibtex
MSC-Numbers: 35P99
Keywords and phrases: Steklov eigenfunctions, nodal sets
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Abstract:
We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in ℝⁿ - the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain Ω is C², we prove a doubling property for the eigenfunction u. We estimate the Hausdorff ℋⁿ⁻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(λ⁶). 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].