Homogenization of the boundary value for the Dirichlet Problem

Details Homogenization of the boundary value for the Dirichlet Problem Homogenization of the boundary value for the Dirichlet Problem

2012-01-31

arxiv.org/abs/1201.6683v1

Mathematics

Analysis of PDEs

Scientific paper

The Dirichlet problem with oscillating boundary data is the subject of study in this paper. It turns out that due to integral representation of such problems we can reduce the study to the case of surface integrals of rapidly oscillating functions, and their limit behavior: $$ \lim_{\e \to 0} \int_{\Gamma} g(y,\frac{y}{\e}) d\sigma_y, $$ where $g(x,y)$, represents the boundary value in the Dirichlet problem. The lower dimensional character of the surface $\Gamma$ produces unexpected and surprising effective limits. In general, the limit of the integral depends strongly on the sequence $\e=\e_j$ chosen. Notwithstanding this, when the surface does not have flat portions with \emph{rational directions} a full averaging takes place and we obtain a unique effective limit in the above integral. The results here are connected to recent works of D. G\'erard-Varet and N. Masmoudi, where they study this problem in combination with homogenization of the operator, in convex domains.

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