Partial Regularity in Nonlocal Problems

A lecture by Giuseppe Mingione explores the theory of partial regular regularity for elliptic systems in nonlocal problems, presented at the Workshop on "Degenerate and Singular PDEs" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in February 2025. Discover how this theory replaces the classical De Giorgi-Nash-Moser approach for scalar equations by establishing that solutions are regular outside a negligible closed subset known as the singular set. Learn about Hausdorff dimension estimates on these generally non-empty singular sets and explore the extension of classical, local partial regularity theory to nonlinear integrodifferential systems. The 51-minute talk presents fundamental tools for proving epsilon-regularity theorems in general non-local settings, drawing from recent joint work with Cristiana De Filippis (Parma) and Simon Nowak (Bielefeld), and builds upon the classical foundations established by Giusti & Miranda and Morrey, which themselves were inspired by De Giorgi's seminal ideas for minimal surfaces.