Homogenization and error estimates of free boundary velocities in periodic media

Abstract

In this note I describe a recent result ([14]- [15]) on homogenization and error estimates of a free boundary problem, which describes quasi-static contact angle dynamics on inhomogeneous surface. The method presented here also applies to more general class of free boundary problems with oscillating boundary velocities.Let us define e i ∈ IR n , i = 1, ..., n such that e 1 = (1, 0, .., 0), e 2 = (0, 1, 0, .., 0), ..., and e n = (0, ..., 0, 1), and consider a Lipschitz continuous functionwith Lipschitz constant L. In this paper we are interested in the behavior, as → 0, of the viscosity solutions u ≥ 0 of the following problem with K = {|x| ≤ 1} and with initial data u 0 :We refer to Γ t (u ) := ∂{u (·, t) > 0} − ∂K as the free boundary of u at time t. Note that if the free boundary of u is smooth, then the boundary moves with outward normal velocity V = u t |Du | , and therefore the second equation in (P ) implies thatwhere ν = ν x,t denotes the outward normal vector at x ∈ Γ t (u) with respect to Ω t (u).(P ) is a simplified model to describe contact line dynamics of quasi-static liquid droplets on an irregular surface ([11]). Here u(x, t) denotes the height of the droplet. Heterogeneities on the surface, represented by g( x ), result in contact lines with a fine scale structure that may lead to pinning of the interface and hysteresis of the overall fluid shape.The following is a summary of the main result in [14]: 1