UNIFORM CONVEXITY AND VARIATIONAL

. Let Ω be a domain in R d . We establish the uniform convexity of the Γ-limit of a sequence of Carath´eodory integrands f ( x,ξ ): Ω × R d → R subjected to a two-sided power-law estimate of coercivity and growth with respect to ξ with exponents α and β , 1 < α ≤ β < ∞ , and having a common modulus of convexity with respect to ξ . In particular, the Γ-limit of a sequence of power-law integrands of the form | ξ | p ( x ) , where the variable exponent p : Ω → [ α,β ] is a measurable function, is uniformly convex. We prove that one can assign a uniformly convex Orlicz space to the Γ-limit of a sequence of power-law integrands. A natural Γ-closed extension of the class of power-law integrands is found. Applications to the homogenization theory for functionals of the calculus of variations and for monotone operators are given.