Measure upper bounds of nodal sets of solutions to Dirichlet problem of Schrödinger 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} Δu+Vu=0\quad \mbox{in}\ Ω,\\[2mm] u=0\quad \mbox{on}\ \partialΩ, \end{array}\right. \end{equation*} where $V\in W^{1,\infty}(Ω)$ is a potential function, and $Ω\subset \mathbb{R}^n$ ($n \geq 2$) is a bounded domain whose boundary is of class $C^{1,α}$ for any $0<α<1$. 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 $Ω$ is $$C\Big(1+\log\left(\|\nabla V\|_{L^{\infty}(Ω)}+1\right)\Big)\cdot\left(\|V\|_{L^{\infty}(Ω)}^{\frac{1}{2}}+\|\nabla V\|_{L^{\infty}(Ω)}^{\frac{1}{2}}+1\right),$$ provided $V$ is analytic, here $C$ is a positive constant depending only on $n$ and $Ω$. In particular, if $\|\nabla V\|_{L^{\infty}(Ω)}$ is small, the upper bound for the measure of the nodal set of $u$ is $C\left(\|V\|^{\frac{1}{2}}_{L^{\infty}(Ω)}+1\right)$, which is sharp in the sense of a famous conjecture of Yau.