A useful underestimate for the convergence of integral functionals

TL;DR

This paper investigates the convergence properties of integral functionals, establishing bounds and conditions for lower semicontinuity and subdifferentiability using the weak lower epi-limit and Ioffe's criterion.

Contribution

It provides new bounds and conditions for the lower semicontinuity and subdifferentiability of integral functionals on general spaces, especially Lebesgue spaces.

Findings

Bounded below by the Fenchel-Moreau-Rockafellar biconjugate of the integrand.

New necessary and sufficient conditions for lower semicontinuity.

Characterization of subdifferentiability in Lebesgue spaces.

Abstract

This article deals with the lower compactness property of a sequence of integrands and the use of this key notion in various domains: convergence theory, optimal control, non-smooth analysis. First about the interchange of the weak epi-limit and the symbol of integration for a sequence of integral functionals. These functionals are defined on a topological space $(\mathcal{X, T})$ where $\mathcal{X}$ is a subset of measurable functions and the $\mathcal{T}$ convergence is stronger than or equal to the convergence in the Bitting sense. Given a sequence $(f_{n})_n$ of integrands, if the integrand $f$ is the weak lower sequential epi-limit of the integrands $f_n$ one of the main results of this article asserts that under the Ioffe's criterion, the $\mathcal{T}$-lower sequential epi-limit of the sequence of integral functionals at the point $x$ is bounded below by the value of the integral functional associated to the Fenchel-Moreau-Rockafellar biconjugate of $f$ at the point $x$. Then the strong-weak semicontinuity (respectively the subdifferentiability) are studied in relation with the Ioffe's criterion. This permits with original proofs to give new conditions for the strong-weak lower semicontinuity at a given point, and to obtain necessary and sufficient conditions for the Fr\'echet and the (weak)Hadamard subdifferentiability of integral functionals on general spaces, particularly on Lebesgue spaces.