Principal frequency of the $p$-Laplacian and the inradius of Euclidean domains

TL;DR

This paper establishes new lower bounds for the principal frequency of the $p$-Laplacian in Euclidean domains, relating it to the inradius, and extends classical eigenvalue bounds to higher dimensions and different $p$ ranges.

Contribution

It provides novel lower bounds for the $p$-Laplacian's first eigenvalue based on inradius, generalizing classical results to higher dimensions and broader $p$ ranges.

Findings

Lower bounds for $p$-Laplacian eigenvalues in terms of inradius for $p>N$.

Extension of bounds to cases with connected boundary for $p > N-1$.

Generalization of classical planar eigenvalue bounds to higher dimensions.

Abstract

We study the lower bounds for the principal frequency of the $p$-Laplacian on $N$-dimensional Euclidean domains. For $p>N$, we obtain a lower bound for the first eigenvalue of the $p$-Laplacian in terms of its inradius, without any assumptions on the topology of the domain. Moreover, we show that a similar lower bound can be obtained if $p > N-1$ assuming the boundary is connected. This result can be viewed as a generalization of the classical bounds for the first eigenvalue of the Laplace operator on simply connected planar domains.