Homogenization and error estimates of free boundary velocities in periodic media

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 ei ∈ IR, i = 1, ..., n such that e1 = (1, 0, .., 0), e2 = (0, 1, 0, .., 0), ..., and en = (0, ..., 0, 1), and consider a Lipschitz continuous function g : IR → [m,M ], g(x+ ei) = g(x) for i = 1, ..., n with 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 u0: (P )  −∆xu (·, t) = 0 in {u > 0} −K u t = |Dxu |(|Dxu | − g( )) on ∂{u > 0}