A sharp lower bound for the nodal volume of harmonic functions

Abstract

Let $u$ be a non-zero real-valued harmonic function in $B_4\subset {\mathbb{R}}^n$ with $n\ge 3$ and $u(0)=0$. We prove that $${\mathcal{H}}^{n-1}(\{u(x)=0\}\cap B_2)\ge C_n\mathcal{N},$$ where $C_n$ is a positive constant depending only on $n$, and $\mathcal{N}$ is the doubling index defined by $$\mathcal{N}={\log }_2\frac{{\sup }_{B_1}|u|}{{\sup }_{B_{\frac{1}{2}}}|u|}.$$ The linear dependence on $\mathcal{N}$ is optimal. This estimate confirms a folklore conjecture on the nodal volume of harmonic functions.