Preprint 19/2014

Nodal Sets of Steklov Eigenfunctions

Katarina Bellova and Fanghua Lin

Contact the author: Please use for correspondence this email.
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
MSC-Numbers: 35P99
Keywords and phrases: Steklov eigenfunctions, nodal sets
Download full preprint: PDF (567 kB)

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in n - 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 n2-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].

03.07.2017, 01:42