Lyapunov-type Inequalities for Partial Differential Equations

TL;DR

This paper develops Lyapunov inequalities for elliptic PDEs, including singular and degenerate cases, providing new bounds for the first eigenvalue of the p-Laplacian and extending existing theoretical results.

Contribution

It introduces Lyapunov inequalities for a broad class of elliptic operators, including singular and degenerate problems, and derives novel lower bounds for the first eigenvalue.

Findings

Derived Lyapunov inequalities for elliptic operators

Established lower bounds for the first eigenvalue of the p-Laplacian

Compared new bounds with existing literature

Abstract

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the $p-$Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the $p-$Laplacian, and compare them with the usual ones in the literature.