Gamma-Convergence of Convex Functions, Conjugates, and Subdifferentials

We generalize the duality principle for the $Γ$-convergence of proper convex lower semicontinuous functions, which was previously known only in separable reflexive Banach spaces. Aditionally, We extend this duality to the broader class of weakly compactly generated (WCG) Banach spaces. Under a suitable hypotheses, we show that $Γ$-convergence in the strong topology is equivalent to $Γ$-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 extension to the WCG framework allows the inclusion of every separable space and every reflexive space, as well as important non-reflexive that might not be separable, such as $L^{1}(μ)$ for an arbitrary $σ$-finite measure. As an application, we derive dual characterizations of the $Γ$-convergence of convex integral functionals on $L^p$ spaces. The proof of the main result relies on a separable reduction approach in WCG spaces using modern tools of Banach space theory, which, as a byproduct, provides new insight into the internal structure of WCG Banach spaces, in a spirit closely related to the structural phenomena underlying the theory of Asplund spaces [Cúth and Fabian (2016)].