Partial regularity in nonlocal systems I

TL;DR

This paper establishes partial regularity results for solutions to nonlinear nonlocal integro-differential systems, identifying a small singular set with explicitly bounded Hausdorff dimension using energy thresholds and potential theory.

Contribution

It introduces a novel approach combining energy thresholds and nonlinear potentials to analyze regularity in nonlocal systems, providing explicit bounds on the singular set.

Findings

Solutions are regular outside a negligible set with bounded Hausdorff dimension.

Quantitative energy thresholds characterize the singular set.

Fine properties of solutions are derived under sharp assumptions on data and kernels.

Abstract

Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy thresholds for nonlocal excess functionals. The analysis is carried out via the use of nonlinear potentials and allows to derive fine properties of solutions under sharp assumptions on data and kernel coefficients.