Homogenization of the boundary value for the Dirichlet Problem

Sunghan Kim, Ki-ahm Lee, Henrik Shahgholian

Published 2012-01-31, updated 2017-01-14Version 2

In this paper,we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals, which has been a well-known yet unsolved problem. Based on ergodic theory, we find a sharp geometric condition called IDDC under which the averaging takes place. It should be stressed that this condition does not imply any control on the curvature of the given surface. As an application to partial differential equations, we prove boundary layer homogenization for elliptic systems with Dirichlet boundary data, by characterizing the oscillatory Poisson kernels in a periodic manner.