2017/04/11 by Annalisa Cesaroni, Serena Dipierro, Cesaroni, Annalisa +5 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP
paper · pdf · doi:10.48550/arxiv.1704.03195
To appear in Calc. Var. Partial Differential Equations
openalex publication_date 2017/04/11 · arxiv created 2018/03/03 · arxiv updated 2018/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 different behaviors of the energy at different 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 classification 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.