Homogenization of Oscillating Free Boundaries: The Elliptic Case

Abstract

We study the smallest viscosity solution of 1-phase free boundary problem of flame propagation type with a highly oscillating gradient condition on the free boundary. where Ω(u ϵ) = {u ϵ > 0} and f(x + e) = f(x) for e ∈ ℤ n and f(x 1,…, −x i ,…, x n ) = f(x 1,…, x i ,…, x n ). The existence of solutions, u ϵ, and their uniform gradient estimates let us extract convergent subsequences with a limit function u. We show the existence of plane-like global solutions called “correctors” that have the largest slope, α(ν), in each direction ν ∈ S n−1, for any dimension, and we also prove that the correctors will be trapped by two parallel planes having a width of a size of order ϵ. When D is a ball and n = 2, we are able to show that a flat spot occurs when α(ν o ) > lim supν→ν o α(ν) for the unit normal vector ν o to a free boundary ∂Ω(u) at a point x o ∈ ∂Ω(u) and when ∂Ω (u) is locally convex at x o , while the limit of the corresponding energy minimizer has a circular free boundary. The homogenized free boundary condition satisfied by u is analyzed.