Nodal Sets of Steklov Eigenfunctions

2014/02/18 by Bellova, Katarina, Lin, Fanghua
#35P99 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1402.4323

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 \mathcal 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(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].

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