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

We investigate the interior nodal sets Nλ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be Cλ. The singular sets Sλ consist of finitely many points on the nodal sets. We are able to prove that the Hausdorff measure H0(Sλ) is at most Cλ2. Furthermore, we obtain an upper bound for the measure of interior nodal sets, H1(Nλ) Cλ32. Here the positive constants C depend only on the surfaces.

nodal sets, upper bound, Steklov eigenfunctions
Mathematical Subject Classification 2010
Primary: 35P15, 35P20, 58C40, 28A78
Received: 8 July 2015
Revised: 20 December 2015
Accepted: 26 February 2016
Published: 3 July 2016
Jiuyi Zhu
Department of Mathematics
Johns Hopkins University
313 Krieger Hall
3400 N. Charles Street
Baltimore, MD 21218
United States