Almost minimizers for certain fractional variational problems

TL;DR

This paper introduces the concept of almost minimizers for fractional variational problems, demonstrating their regularity properties and extending the understanding of fractional harmonic functions and obstacle problems.

Contribution

It defines almost minimizers for fractional variational problems and establishes their regularity, advancing the analysis of fractional harmonic functions and obstacle problems.

Findings

Almost minimizers are almost Lipschitz or $C^{1,eta}$-regular.

Regularity results depend on the range of parameters.

The approach uses the Caffarelli-Silvestre extension.

Abstract

In this paper we introduce a notion of almost minimizers for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli-Silvestre extension. In particular, we study almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle. We show that for a certain range of parameters, almost minimizers are almost Lipschitz or $C^{1,\beta}$-regular.