Fractional De Giorgi classes and applications to nonlocal regularity theory

TL;DR

This paper reviews recent advances in the regularity of solutions to nonlocal variational problems, focusing on fractional De Giorgi classes and their significance in nonlocal regularity theory.

Contribution

It introduces the concept of fractional De Giorgi classes, discusses their role in nonlocal regularity, and outlines open questions in the field.

Findings

Fractional De Giorgi classes are crucial for nonlocal regularity analysis.

Recent results improve understanding of solution regularity in nonlocal problems.

Open questions highlight future research directions.

Abstract

We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and propose some open questions in the subject.