积分泛函收敛性的一个有用的估计
A useful underestimate for the convergence of integral functionals
摘要：
本文讨论了被积函数序列的下紧性，以及这个关键概念在收敛理论、最优控制、非光滑分析等领域的应用。第一，积分泛函列的弱上极限与积分符号的互换。这些泛函被定义在拓扑空间$(\mathcal{X, T})$上，其中$\mathcal{X}$是可测函数的子集，并且$\mathcal{T}$收敛强于或等于Bitting意义上的收敛。给定被积函数序列$(f_{n})_n$，若被积函数$f$是被积函数序列的弱序列上极限$f_n$本文的主要结果之一是在Ioffe准则下，积分泛函序列在点$x$处的$\mathcal{T}$ -序列上下限由与Fenchel-Moreau相关的积分泛函的值所限定，$x$点处的$f$的Rockafellar双共轭物。然后研究了与Ioffe判据有关的强-弱连续性（次可微性）。这允许与原来的证明给出新的条件的强弱下连续在一个给定的点，并获得充要条件的Fréchet和（弱）Hadamard次微分的积分泛函在一般空间，特别是勒贝格空间。
英文摘要：
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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