Bounds on the Principal Frequency of the $p$-Laplacian

TL;DR

This paper extends classical bounds on the principal frequency of the Laplacian to the p-Laplacian, providing new lower bounds related to domain size and eigenfunction nodal sets.

Contribution

It introduces lower bounds for the p-Laplacian's principal frequency, generalizing results from the classical Laplace case to p≠2.

Findings

Derived lower bounds involving the inner radius for the p-Laplacian

Extended classical eigenvalue bounds to the p-Laplacian case

Provided bounds on the size of nodal sets of eigenfunctions

Abstract

This paper is concerned with the lower bounds for the principal frequency of the $p$-Laplacian on $n$-dimensional Euclidean domains. In particular, we extend the classical results involving the inner radius of a domain and the first eigenvalue of the Laplace operator to the case $p\neq2$. As a by-product, we obtain a lower bound on the size of the nodal set of an eigenfunction of the $p$-Laplacian on planar domains.