In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e., \begin{ equation*} { \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{ equation*} where the domain $\Omega\subseteq\mathbb{ R}^n$, $u_{ \nu}$ means the derivative of $u$ along the outer normal direction of $\partial\Omega$. We 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 $C(\sqrt{ \lambda}+\mu^{ \alpha}+\mu^{ -c\alpha})$ for some positive number $\alpha$, where $c$ is a positive constant depending only on $n$, and $C$ is a positive constant depending on $n$, $\Omega$, $\Gamma$ and $\alpha$.
Categories: math.AP