Quantitative Homogenization and Large-Scale Regularity

Homogenization is a fundamental tool for describing the effective behavior of partial differential equations with highly oscillatory coefficients and has attracted significant attention for several decades. In particular, beyond qualitative convergence, homogenization provides a framework for transferring large-scale regularity of the coefficient field to solutions in a quantitative manner. In this talk, I will explain how large-scale properties of the coefficients can compensate for the absence of classical regularity assumptions, such as uniform ellipticity and the Dahlberg–Kenig–Pipher condition arising in the study of the Dirichlet problem. These results illustrate how quantitative homogenization techniques yield robust regularity estimates in settings with limited microscopic control.