Almost sharp lower bound for the nodal volume of harmonic functions

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Speaker

Lakshmi Priya M.E. (Tel Aviv)

Date & Time

Mon, 30 October 2023, 14:00 to 14:50

Venue

Chern Lecture Hall & Online

Resources

Abstract

In this talk, I will discuss the relation between the growth of harmonic functions and their nodal volume. Let $\mathbb{R}^n \rightarrow \mathbb{R}$ 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

$\sup_{B(0,1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|.$

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

Zoom Link: https://us02web.zoom.us/j/88670406480
Meeting ID: 886 7040 6480