Measure upper bounds of nodal sets of Robin eigenfunctions

Abstract

In this paper, we will establish the upper bounds of the Hausdorff measure of nodal sets of eigenfunctions with the Robin boundary conditions, i.e.,

$$\begin{aligned} {\left\{ \begin{array}{l} \triangle u+\lambda u=0,\quad in\quad \Omega ,\\ u_{\nu }+\mu u=0,\quad on\quad \partial \Omega , \end{array} \right. } \end{aligned}$$

where the domain $\Omega \subseteq \mathbb {R}^n$, $u_{\nu }$ is the derivative of u along the outer normal direction on $\partial \Omega$. We will show that, if $\Omega$ is bounded and analytic, and the corresponding eigenvalue $\lambda$ is large enough, then the measure upper bounds for the nodal sets of eigenfunctions are $C\sqrt{\lambda }$, where C is a positive constant depending only on n and $\Omega$ but not on $\mu$. We also show that, if $\partial \Omega$ is $C^{\infty }$ smooth and $\partial \Omega {\setminus }\Gamma$ is piecewise analytic, where $\Gamma \subseteq \partial \Omega$ is a union of some $n-2$ dimensional submanifolds of $\partial \Omega$, $\mu >0$, and $\lambda$ is large enough, then the corresponding measure upper bounds for the nodal sets of u are $C\left( \sqrt{\lambda }+\mu ^{\alpha }+\mu ^{-c\alpha }\right)$ for any $\alpha \in (0,1)$, where C is a positive constant depending on $\alpha$, n, $\Omega$ and $\Gamma$, and c is a positive constant depending only on n.