A lower bound for the principal eigenvalue of fully nonlinear elliptic operators

TL;DR

This paper introduces a new technique to establish lower bounds for the principal eigenvalue of fully nonlinear elliptic operators, with applications to p-Laplacian and infinity Laplacian eigenvalues.

Contribution

It develops a novel method to derive lower bounds for the principal eigenvalue of fully nonlinear elliptic operators, demonstrated through several examples.

Findings

Established a lower bound for the principal eigenvalue of fully nonlinear elliptic operators.

Proved the limit of p-Laplacian eigenvalues as p approaches infinity equals the infinity Laplacian eigenvalue.

Applied the method to specific cases, confirming theoretical predictions.

Abstract

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that $\lim_{p\to \infty}\lambda_{1,p}=\lambda_{1,\infty}=\left(\frac{\pi}{2R}\right)^2$ where $\lambda_{1,p}$ and $\lambda_{1,\infty}$ are the principal eigenvalue for the homogeneous $p$-laplacian and the homogeneous infinity laplacian respectively.