Duality for the $\Gamma$-convergence of convex functions

Abstract

We extend the duality principle for the $\Gamma$-convergence of convex lower semicontinuous functions, which was previously established only in separable reflexive Banach spaces, to the broader class of weakly compactly generated (WCG) Banach spaces, addressing a question of Fitzpatrick and Lewis. Under the same classical hypothesis of equicoercivity, we show that $\Gamma$-convergence in the norm topology is equivalent to $\Gamma$-convergence of the Fenchel conjugates in the weak$^\ast$ topology. We further prove that this duality is equivalent to the graphical convergence of the associated subdifferentials with respect to the product topology given by the norm on the primal space and the weak$^\ast$ topology on the dual. The WCG setting encompasses all separable and all reflexive Banach spaces separately, i.e, separable spaces without reflexivity assumptions and reflexive spaces without separability assumptions, as well as important non-reflexive spaces which may fail to be separable, such as $L^1(\mu)$ for an arbitrary $\sigma$-finite measure. As an application, we derive dual characterizations of the $\Gamma$-convergence of convex integral functionals on $L^p$ spaces ($1\leq p<\infty $).