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Journal of Convex Analysis 20 (2013), No. 2, 355--376
Copyright Heldermann Verlag 2013

An Upper Bound for the Convergence of Integral Functionals

Emmanuel Giner
Institut de Mathématiques, Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse 04, France


Given a $\sigma$-finite measure space $(\Omega, \mathcal{T}, \mu)$ endowed with a $\mu$-complete tribe, a separable Banach space $E$, we consider a topological vector space $(X,T)$, $X$ being a decomposable subspace of measurable $E$-valued functions defined on $\Omega$. Under a reasonable assumption on the vector topology $T$, we show that if ${(f_{n})}_{n}$ is a sequence of extended real-valued measurable integrands defined on the product $\Omega\times E$, with upper epi-limit (or upper $\Gamma$-limit) $f=ls_{e} f_{n}$, then $I_f$ is in many cases an upper bound for the $T$-upper epi-limit of the sequence $(I_{f_{n}})_n$, where $I_f$, $I_{f_{n}}$ are the integral functionals defined on $X$ associated to the integrands $f$, $f_n$. The cases of Lebesgue spaces endowed with its strong, weak, or Mackey topologies are reached. We discuss also the necessity of the given conditions.

Keywords: Integral functional, upper epi-limits, $\Gamma$-convergence, finally equi-integrable sets.

MSC: 26A16, 26A24, 26E15, 28B20, 49J52, 54C35

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