Measure upper bounds of nodal sets of solutions to Dirichlet problem of Schr\"{o}dinger equations

Abstract

In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} \Delta u+Vu=0\quad \mbox{in}\ \Omega,\\[2mm] u=0\quad \mbox{on}\ \partial\Omega, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(\Omega)$ is a potential function, and $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded domain whose boundary is of class $C^{1,\alpha}$ for any $0<\alpha<1$. By developing a delicate dividing iteration procedure, we show that upper bound of the $(n-1)$-dimensional Hausdorff measure of the nodal set of $u$ in $\Omega$ is $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(\Omega)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+\|\nabla V\|_{L^{\infty}(\Omega)}^{\frac{1}{2}}+1\right),$$ provided $V$ is analytic, here $C$ is a positive constant depending only on $n$ and $\Omega$. In particular, if $\|\nabla V\|_{L^{\infty}(\Omega)}$ is small, the upper bound for the measure of the nodal set of $u$ is $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(\Omega)}+1\right)$, which is sharp in the sense of a famous conjecture of Yau.