We study the regularity of solutions to a nonlocal version of the image denoising model and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) β-Hölder continuous for some β ∈ (1 − s, 1], where s ∈ (0, 1) is a parameter related to the nonlocal operator, we prove that the solution is also β-Hölder continuous.

We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces. Nevertheless, we show that when the width of the slab is large the minimizers are not flat discs, as it happens in the classical setting, and, in particular, in dimension  2 2 we provide a quantitative bound on the stickiness property exhibited by the minimizers. Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon.

We consider the nonexistence of minimizers for the energy containing a nonlocal perimeter with a general kernel K , a Riesz potential, and a background potential in \mathbb{R}^N with N\geq2 under the volume constraint. We show that the energy has no minimizer for a sufficiently large volume under suitable assumptions on K . The proof is based on the partition of a minimizer and the comparison of the sum of the energy for each part with the energy for the original configuration.

No abstract

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the assumption that the discrepancy measure vanishes on the boundary. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.