Calculus of Variations and Geometric Measure Theory
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A. Cesaroni - S. Dipierro - M. Novaga - E. Valdinoci

Minimizers for nonlocal perimeters of Minkowski type

created by novaga on 11 Apr 2017
modified on 20 Jun 2017


Submitted Paper

Inserted: 11 apr 2017
Last Updated: 20 jun 2017

Year: 2017


We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This problem is related by a generalized coarea formula to a Dirichlet energy functional in which the energy density is the local oscillation of a function.

These two nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the diff erent behavior of the energy at di fferent scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features.

In this paper, we provide a series of results concerning existence, rigidity and classi cation of minimizers, compactness results, isoperimetric inequalities, Poincaré-Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation.


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