Legendre-type integrands and convex integral functions

TL;DR

This paper investigates the properties of integral functionals derived from convex functions on Euclidean spaces, establishing conditions for strong rotundity and analyzing entropy functions like Boltzmann-Shannon and Fermi-Dirac.

Contribution

It provides new sufficient conditions for strong rotundity of integral functionals and demonstrates that important entropy functions satisfy these conditions.

Findings

Boltzmann-Shannon and Fermi-Dirac entropies are strongly rotund.

Established criteria for strong rotundity of integral functionals.

Analyzed convergence in measure and provided counterexamples.

Abstract

In this paper, we study the properties of integral functionals induced on $L^1_E (S,\mu)$ by closed convex functions on a Euclidean space $E$. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We show that in this generality functions such as the Boltzmann-Shannon entropy and the Fermi-Dirac entropy are strongly rotund. We also study convergence in measure and give various limiting counterexample.