A Homogenization Method for non Variational Problems
L. A. Cafferelli · Current Developments in Mathematics · 2004
We would like to discuss in these lectures three problems of homogenization and their interplay.They are:A) The construction of plane-like solutions to the minimal surface equation in periodic media.B) Pulsating wave solutions to a combustion problem and its homogenization limit.C) Existence of homogenization limits for solutions to fully linear equations in ergodic random media.We will try to point out what the main techniques are, and their common aspects. Part 1) The construction of plane-like solutions to periodic minimal surface equations.In two dimensions, minimal surfaces are just geodesics: we are given in R 2 a differential of length a(x, ν) and given two points x, y, we want to minimizeHere s is the usual differential of length, σ the unit tangent vector.The function a(x, σ) is periodic in x, strictly positive (0 < λ ≤ a(x, σ) ≤ Λ) and, to avoid the formation of Young measures (that is: oscillatory zig-zags) when trying to construct geodesics, it must satisfy " |v|a x, v |v| is a strictly convex cone."This is a "classical" condition of ellipticity for area minimizers, and the regularity of solutions has been studied, for instance by Schoen and Simon.