Hausdorff measure of nodal sets of analytic Steklov eigenfunctions

Let $(\Omega, g)$ be a real analytic Riemannian manifold with real analytic boundary $\partial \Omega$. Let $\psi_{\lambda}$ be an eigenfunction of the Dirichlet-to-Neumann operator $\Lambda$ of $(\Omega, g, \partial \Omega)$ of eigenvalue $\lambda$. Let $N_{\lambda}$ be its nodal set. Then, there exists a constant $C \gt 0$ depending only on $(\Omega, g, \partial \Omega)$ so that\[\mathcal{H}^{n-2} (\mathcal{N}_{\lambda}) \leq C \lambda\]This proves a conjecture of F. H. Lin and K. Bellova.

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Accepted 18 June 2015

Published 23 May 2016