Convexity properties of Dirichlet integrals and Picone-type inequalities

Lorenzo Brasco, Giovanni Franzina
2014 Kodai Mathematical Journal  
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. 2010 Mathematics Subject Classification. 47J10, 39B62, 35B50. Key words and phrases. Nonlinear eigenvalue problems; uniqueness of
more » ... iqueness of eigenfunctions; Hardy inequalities; maximum principle; nonlocal equations. 1 This formula is called "identity" even if it is an inequality, because the difference of the two terms can be written as which is indeed non positive. The latter is the equality which appears in the original paper [25] by Mauro Picone, after whom the formula is named. The identity is used there to obtain comparison principles for ordinary differential equations of Sturm-Liouville type.
doi:10.2996/kmj/1414674621 fatcat:b45s5w3dpzbwlbyrr443zuyk34