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APRG Seminar

Title: Almost sharp lower bound for the nodal volume of harmonic functions
Speaker: Lakshmi Priya M.E. (Tel Aviv University, Tel Aviv, Israel)
Date: 14 September 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department

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 \begin{equation} \sup_{B(0,1)}|u| = 2^N \sup_{B(0,\frac{1}{2})}|u|. \end{equation} I will present a result, obtained jointly with A. Logunov and A. Sartori, where we prove an almost sharp result, namely: \begin{equation} \mathcal{H}^{n-1}({u=0} \cap B(0,2)) \gtrsim_{n,\varepsilon} N^{1-\varepsilon}, \end{equation} where $\mathcal{H}^{n-1}$ denotes the $(n-1)$ dimensional Hausdorff measure.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 17 May 2024