Interior nodal sets of Steklov eigenfunctions on surfaces

Zhu, Jiuyi

doi:10.2140/apde.2016.9.859

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Interior nodal sets of Steklov eigenfunctions on surfaces

Jiuyi Zhu

Vol. 9 (2016), No. 4, 859–880

DOI: 10.2140/apde.2016.9.859

Abstract

We investigate the interior nodal sets $N_{λ}$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $C λ$ . The singular sets $S_{λ}$ consist of finitely many points on the nodal sets. We are able to prove that the Hausdorff measure $H^{0} (S_{λ})$ is at most $C λ^{2}$ . Furthermore, we obtain an upper bound for the measure of interior nodal sets, $H^{1} (N_{λ}) \leq C λ^{3 ∕ 2}$ . Here the positive constants $C$ depend only on the surfaces.

Keywords

nodal sets, upper bound, Steklov eigenfunctions

Mathematical Subject Classification 2010

Primary: 35P15, 35P20, 58C40, 28A78

Milestones

Received: 8 July 2015

Revised: 20 December 2015

Accepted: 26 February 2016

Published: 3 July 2016

Authors

Jiuyi Zhu
Department of Mathematics Johns Hopkins University 313 Krieger Hall 3400 N. Charles Street Baltimore, MD 21218 United States

Vol. 9, No. 4, 2016