

Preprint 19/2014
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].