L ^∞ and L ¹ Variations on a Theme of Γ-Convergence

Abstract

The theory of T-convergence is an important tool in Calculus of Variations, because the equicoercivity and the T-convergence of a sequence of functionals F _∊ to F _o imply the weak convergence of minima (u _∊ → u ₀) and the convergence of F _∊(uε) to F ₀(u _o). Unfortunately, in the general case, the T-convergence of F _∊ to F _o do not imply the Γ-convergence of (F _∊ + G) to (F _o + G). Thus, if we want study the convergence of the sequence u _∊, where u _∊ is a minimum of F _∊ over the convex set

$$K(\psi ) = \left\{ {\upsilon \in W_o^{s,p}(\Omega ){\rm{ }}\upsilon \ge \psi {\rm{ a}}{\rm{.e}}{\rm{. in }}\Omega } \right\}$$

we must proof the Γ—convergence of

$$({F_ \in } + \delta ({\rm K}(\psi ))) {\rm{to (}}{F_o} + \delta ({\rm{K(}}\psi {\rm{)))}}{\rm{.}}$$

The point of view we adopt is the minimization of functionals: thus no Euler equation will be really written.