Nonlinear nonlocal equations in Reifenberg flat domains

TL;DR

This paper studies boundary regularity of solutions to nonlinear fractional p-Laplace equations on irregular Reifenberg flat domains, providing new results even for linear cases.

Contribution

It establishes novel boundary regularity results for fractional p-Laplace equations on Reifenberg flat domains, extending beyond Lipschitz domains.

Findings

Solutions are regular near the boundary under Reifenberg flatness.

Gradient regularity results are obtained for solutions.

Results are new even for linear fractional Laplace equations.

Abstract

We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish several fine boundary regularity results for solutions, and their gradient, near the boundary. To the best of our knowledge, each of our results is new even in the linear case.