Homogenization of a model problem on contact angle dynamics

In this paper we consider homogenization of oscillating free boundary velocities in periodic media, with general initial data. We prove that there is a unique and stable effective free boundary velocity in the homogenization limit. 0 Introduction Consider a compact set K ⊂ IR with smooth boundary ∂K. Suppose that a bounded domain Ω contains K and let Ω0 = Ω − K and Γ0 = ∂Ω. We also assume that Int(Ω) = Ω̄. Note that ∂Ω0 = Γ0 ∪ ∂K. For a continuous function f(x, t) : IR × [0,∞) → (0,∞), let u0 satisfy −∆u0 = 0 in Ω0, u0 = f on K, and u0 = 0 on Γ0. (see Figure 1.) Let us define ei ∈ IR, i = 1, ..., n such that (0.2) e1 = (1, 0, ..0), e2 = (0, 1, 0, .., 0), ..., and en = (0, .., 0, 1). Consider a Lipschitz continuous function g : IR → [a, b], g(x+ ei) = g(x) for i = 1, ..., n with Lipschitz constant M . In this paper we consider the behavior, as ǫ → 0, of the nonnegative (viscosity) solutions uǫ ≥ 0 of the following problem