1. Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory
- Author
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Xavier Cabré, Universitat Politècnica de Catalunya. Departament de Matemàtiques, and Universitat Politècnica de Catalunya. EDP - Equacions en Derivades Parcials i Aplicacions
- Subjects
Pure mathematics ,Open set ,Null-Lagrangian ,Space (mathematics) ,53 Differential geometry [Classificació AMS] ,01 natural sciences ,Omega ,Perimeter ,Mathematics - Analysis of PDEs ,Nonlocal minimal surfaces ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,49 Calculus of variations and optimal control ,optimization [Classificació AMS] ,0101 mathematics ,Mathematics ,Mean curvature ,Applied Mathematics ,010102 general mathematics ,Null (mathematics) ,Matemàtiques i estadística [Àrees temàtiques de la UPC] ,Nonlocal perimeter ,Foliation ,49 Calculus of variations and optimal control [Classificació AMS] ,Calibration ,Viscosity solutions ,010307 mathematical physics ,optimization ,Analysis of PDEs (math.AP) - Abstract
For nonnegative even kernels $K$, we consider the $K$-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated $K$-nonlocal mean curvature equation in an open set $\Omega\subset\mathbb{R}^n$, we built a calibration for the nonlocal perimeter inside $\Omega\subset\mathbb{R}^n$. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in $\Omega$ of each leaf of the foliation. As an application, we prove the minimality of $K$-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions., Comment: To appear in Annali di Matematica Pura ed Applicata
- Published
- 2020
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