Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes

TL;DR

This paper extends classical regularity results and Harnack inequalities to nonlocal fractional problems by introducing fractional De Giorgi classes, demonstrating that minimizers are locally bounded and Hölder continuous.

Contribution

It introduces a unified approach using fractional De Giorgi classes to establish regularity and Harnack inequalities for nonlocal problems, extending classical results to the fractional setting.

Findings

Minimizers are locally bounded and Hölder continuous.

Solutions satisfy a Harnack inequality.

Applicable to general nonlinear integral equations.

Abstract

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, H\"older continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations.