An Introduction to Homogenization and Gamma-convergence

CONTENTS 1. Γ-convergence for integral functionals. 2. A general compactness result. 3. Homogenization formulas. 4. Examples: homogenization without standard growth conditions. 5. Examples: other homogenization formulas. 43 44 Andrea Braides This paper contains the abstract of five lectures conceived as an introduction to Γ-convergence methods in the theory of Homogenization, and delivered on September 8–10, 1993 as part of the " School on Homogenization " at the ICTP, Trieste. Its content is strictly linked and complementary to the subject of the courses held at the same School by A. Defranceschi and G. Buttazzo. Prerequisites are some basic knowledge of functional analysis and of Sobolev spaces (as a reference we shall use the books by Adams [3] and Ziemer [29]; see also the Appendix to the Lecture Notes by A. Defranceschi in this volume). A list of notations can be found at the end of this paper. The subject of these lectures is the study of the asymptotic behaviour as ε goes to 0 of integral functionals of the form (1.1) F ε (u) = Ω f (x ε , Du(x)) dx, defined on some (subset of a) Sobolev space W 1,p (Ω; IR N) (in general, of vector-valued functions), when f = f (y, ξ) is a Borel function, (almost) periodic in the variable y, and satisfying the so-called " natural growth " conditions with respect to the variable ξ. Integrals of this form model various phenomena in Mathematical Physics in the presence of microstructures (the vanishing parameter ε gives the microscopic scale of the media). The function f represents the energy density at this lower scale. As an example we can think of u representing a deformation, and F ε being the stored energy of a cellular elastic material with Ω as a reference configuration. In other applications u is a difference of potential in a condenser composed of periodically distributed material, occupying the region Ω, etc. The main question we are going to answer is: does the (medium modeled by the) energy F ε behave as a homogeneous medium in the limit? (and if so: can we say something about this homogeneous limit?) First we have to give a precise meaning to this statement. The behaviour of the media described by the integral in (1.1) is given by the behaviour of boundary value problems of the Calculus of Variations of the form (1.2) min Ω f (x ε …