Local H\"older regularity of minimizers for nonlocal variational problems

TL;DR

This paper proves that in two dimensions, solutions to a nonlocal variational problem related to image denoising inherit the same H"older regularity as the input data, under certain conditions.

Contribution

It establishes the regularity transfer from data to minimizers for a class of nonlocal variational problems in two dimensions.

Findings

Minimizers are H"older continuous if the data is H"older continuous.

Regularity of solutions matches the regularity of the original image.

Results are specific to two-dimensional cases.

Abstract

We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same H\"older regularity as the original image. More precisely, if the datum is (locally) $\beta$-H\"older continuous for some $\beta \in (1-s,\,1]$, where $s \in (0,1)$ is a parameter related to the nonlocal operator, we prove that the solution is also $\beta$-H\"older continuous.