Abstract :

In this talk, I will discuss the relation between the growth of harmonic functions and their nodal volume. Let $u:\mathbb{R}^n \rightarrow \mathbb{R}$ be a harmonic function, where $n\geq 2$. One way to quantify the growth of $u$ in the ball $B(0,1) \subset \mathbb{R}^n$ is via the doubling index $N$, defined by
$\displaystyle\sup_{B(0,1)}|u| = 2^N \displaystyle\sup_{B(0,\frac{1}{2})}|u|$.
I will present a result, obtained jointly with A. Logunov and A. Sartori, where we prove an almost sharp result, namely:
$\mathcal{H}^{n-1}(\{u=0\} \cap B(0,2)) \gtrsim_{n,\varepsilon} N^{1-\varepsilon}$,
where $\mathcal{H}^{n-1}$ denotes the $(n-1)$ dimensional Hausdorff measure.