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MiS Preprint

Nodal Sets of Steklov Eigenfunctions

Katarina Bellova and Fanghua Lin


We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in $\mathbb{R}^n$ - the eigenfunctions of the Dirichlet-to-Neumann map.

Under the assumption that the domain $\Omega$ is $C^2$, we prove a doubling property for the eigenfunction $u$. We estimate the Hausdorff $\mathcal H^{n-2}$-measure of the nodal set of $u|_{\partial \Omega}$ in terms of the eigenvalue $\lambda$ as $\lambda$ grows to infinity. In case that the domain $\Omega$ is analytic, we prove a polynomial bound O($\lambda^6$).

Our arguments, which make heavy use of Almgren's frequency functions, are built on the previous works [Garofalo and Lin, CPAM 40 (1987), no. 3; Lin, CPAM 42 (1989), no. 6].

MSC Codes:
Steklov eigenfunctions, nodal sets

Related publications

2015 Repository Open Access
Katarína Bellová and Fanghua Lin

Nodal sets of Steklov eigenfunctions

In: Calculus of variations and partial differential equations, 54 (2015) 2, pp. 2239-2268