Will Feldman: Introductory program: Homogenization of Oscillating Boundary Data

We will discuss the averaging behavior of fully nonlinear elliptic equations with periodic and random Dirichlet boundary data in general domains. In the periodic case the main difficulty has to do with restricting periodic functions in the whole space to hyperplanes. The action of the hyperplane on the underlying probability space (the torus) is only ergodic when the normal direction is irrational. The resulting homogenized boundary condition may be discontinuous but we will show that homogenization holds anyway as long as the Hausdorff dimension of the discontinuity set is small enough. In the random case the well established obstacle problem method for finding a subadditive quantity has so far been unsuccessful. Instead we will show an approach based on concentration of measure results. This yields a proof of homogenization with algebraic rate in probability. This is partially based on a joint work with Inwon Kim and Takis Souganidis.