Convexity Properties of Dirichlet Integrals and Picone-type Inequalities

TL;DR

This paper explores convexity principles related to Dirichlet integrals and Picone inequalities, generalizing them and applying these results to eigenvalue problems and inequalities in nonlinear and nonlocal contexts.

Contribution

It introduces new generalizations and equivalences of convexity principles for variational integrals, with applications to eigenvalue problems and Hardy inequalities.

Findings

Generalized convexity principles for local and nonlocal integrals

Established equivalences among convexity principles

Applied results to nonlinear eigenvalue problems and Hardy inequalities

Abstract

We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions.