Mathematical Research Letters

Volume 22 (2015)

Number 4

A lower bound for the nodal sets of Steklov eigenfunctions

Pages: 1243 – 1253

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n4.a14

Authors

Xing Wang (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Jiuyi Zhu (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Abstract

We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary—the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_{\lambda}$ be its nodal set. Assume that zero is a regular value of Steklov eigenfunctions. We show that\[H^{n-1} (N_{\lambda}) \geq {C \lambda}^{\frac{3-n}{2}}\]for some positive constant C depending only on the manifold.

Keywords

nodal sets, lower bound, Dirichlet-to-Neumann map, Steklov eigenfunctions

2010 Mathematics Subject Classification

28A78, 35P15, 35R01, 58C40

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Published 24 July 2015