IUMJ

Title: Viscosity method for homogenization of highly oscillating obstacles

Authors: Luis Caffarelli and Ki-ahm Lee

Issue: Volume 57 (2008), Issue 4, 1715-1742

Abstract:

In this paper, we develop a viscosity method for the obstacle problem for highly oscillating obstacles. The least viscosity super solutions $u_{\epsilon}$ of Laplace equation above highly oscillating obstacles are considered. For simplicity, we consider obstacles that are consisted of cylindrical columns distributed periodically. If the decay rate of the capacity of columns is too high or too small, the limit of $u_{\epsilon}$ ends up with trivial solutions. The critical decay rates of having nontrivial solution are obtained with construction of barriers. We also show the limit of $u_{\epsilon}$ satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles in viscosity sense. This method can be extended to a fully nonlinear elliptic equation $F(D^2u)$ homogenous with degree one with the study of a capacity potential and of a point singularity of the given nonlinear equation.