A useful underestimate for the convergence of integral functionals

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 Fréchet and the (weak)Hadamard subdifferentiability of integral functionals on general spaces, particularly on Lebesgue spaces.

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