On the Homogenization of the Oscillating Dirichlet Problem

In this talk we will be interested in the homogenization of the oscillating Dirichlet problem for linear elliptic systems with periodic coefficients. Curiosity about this problem finds roots within the study of the homogenization of the Dirichlet problem with non-oscillating boundary data; Here, there is a boundary layer phenomenon as the oscillating term in the two-scale expansion induces strong gradients at the boundary. As is described by Allaire and Amar, in order to refine the two-scale expansion to yield the expected asymptotic estimates up to the boundary, it is necessary to study the homogenization of the oscillating Dirichlet problem. To treat the oscillating Dirichlet problem one tries to approximate the solution by functions solving Dirichlet problems on half- planes which approximate the boundary. This procedure is very sensitive to the geometry of the domain as the approximating hyperplanes, depending on their normal, may ruin the periodicity of the coefficients.

Our main goal will be to discuss some recent results of Armstrong, Kuusi, Mourrat, and Prange that are contained within their paper, ``Quantitative Analysis of Boundary Layers in Periodic Homogenization.'' In the case of smooth, bounded, uniformly convex domains and with a smoothness assumption on the coefficients and the boundary data, for $p \in \left[2, \infty \right)$ they obtain $L^p$-convergence rates for the homogenization of the oscillating Dirichlet problem for $d \geq 2$ that are optimal for $d\geq3$ and also obtain results on the regularity of the homogenized boundary data.