A useful underestimate for the convergence of integral functionals
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 (X, T) where X is a subset of measurable functions and the T convergence is stronger than or equal to the convergence in the Bitting sense. Given a sequence (fn)n of integrands, if the integrand f is the weak lower sequential epi-limit of the integrands fn one of the main results of this article asserts that under the Ioffe's criterion, the 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.