A Lyapunov type Inequality for Indefinite Weights and Eigenvalue Homogenization

TL;DR

This paper establishes a Lyapunov type inequality for quasilinear problems with indefinite weights, providing bounds on the first eigenvalue and applying these results to eigenvalue homogenization problems with indefinite weights.

Contribution

It introduces a novel Lyapunov inequality that bounds the first eigenvalue using the integral of the weight, applicable to indefinite weights and homogenization.

Findings

First eigenvalue is bounded below by the integral of the weight.

The inequality applies to quasilinear problems with indefinite weights.

Effective bounds are derived for eigenvalue homogenization problems.

Abstract

In this paper we prove a Lyapunov type inequality for quasilinear problems with indefinite weights. We show that the first eigenvalue is bounded below in terms of the integral of the weight, instead of the integral of its positive part. We apply this inequality to some eigenvalue homogenization problems with indefinite weights.