Calculus of Variations and Geometric Measure Theory

L. Brasco - G. Franzina

Convexity properties of Dirichlet integrals and Picone-type inequalities

created by brasco on 28 Feb 2014
modified by franzina on 03 Nov 2015


Published Paper

Inserted: 28 feb 2014
Last Updated: 3 nov 2015

Journal: Kodai Math. J.
Year: 2014

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.


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.

Keywords: Nonlinear eigenvalue problems, Maximum Principle, Uniqueness of eigenfunctions, Hardy inequalities, nonlocal equations