Periodic homogenization and harmonic measures
Guy David, Antoine Gloria, Svitlana Mayboroda, Siguang Qi
TL;DR
The paper addresses when elliptic measures remain well-behaved for operators with rapidly oscillating coefficients in the half-space. It replaces the traditional DKP control with periodic homogenization at small scales, showing that if the homogenized problem satisfies a Carleson-energy bound and the microscale periods decay as $\varepsilon_{kj}\lesssim 2^{\alpha(p)k}$ with $\alpha(p)=(3p-1)/(2(p-1))$, then the solution $u$ satisfies a Carleson bound for $t|\nabla u|^2$, implying favorable elliptic-measure properties. The approach relies on a localized two-scale expansion near the boundary, constructing $u^{2s}=\bar u+\sum 2^{k}\varepsilon_{kj}\chi_{kj}\phi_{kj}^i(\cdot/(2^{k}\varepsilon_{kj}))\partial_i\bar u$ and controlling homogenization errors via a carefully estimated source term; this yields a sharp link between homogenization and boundary regularity. Corollaries show that when the homogenized matrix $\bar A$ is laminate or constant, the required small-scale oscillations can be weakened further, broadening the applicability of the method to classes of rapidly oscillating but homogenizable coefficients.
Abstract
Since the seminal work of Kenig and Pipher, the Dahlberg-Kenig-Pipher (DKP) condition on oscillations of the coefficient matrix became a standard threshold in the study of absolute continuity of the harmonic measure with respect to the Hausdorff measure on the boundary. It has been proved sufficient for absolute continuity in the domains with increasingly complex geometry, and known counterexamples show that in a certain sense it is necessary as well. In the present note, we introduce into the subject ideas from homogenization theory to exhibit a new class of operators for which the elliptic measure is well-behaved, featuring the coefficients violating the DKP condition, and on the contrary, oscillating so quickly, that the homogenization takes place.