Convexity Properties of Dirichlet Integrals and Picone-type 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.

This paper has been written for possible publication in a special volume dedicated to the conference "Third Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's", organized in Tokyo in August 2013

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