Vol. 9, No. 4, 2016

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Interior nodal sets of Steklov eigenfunctions on surfaces

Jiuyi Zhu

Vol. 9 (2016), No. 4, 859–880
Abstract

We investigate the interior nodal sets ${\mathsc{N}}_{\lambda }$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $C\lambda$. The singular sets ${\mathsc{S}}_{\lambda }$ consist of finitely many points on the nodal sets. We are able to prove that the Hausdorff measure ${H}^{0}\left({\mathsc{S}}_{\lambda }\right)$ is at most $C{\lambda }^{2}$. Furthermore, we obtain an upper bound for the measure of interior nodal sets, ${H}^{1}\left({\mathsc{N}}_{\lambda }\right)\le C{\lambda }^{3∕2}$. 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