Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory

• Published in: Annali di matematica pura ed applicata Vol. 199; no. 5; pp. 1979 - 1995
• Main Author: Cabré, Xavier
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2020; Springer Nature B.V
• Subjects: Calibration; Mathematics; Mathematics and Statistics; Viscosity; Null-Lagrangian; 53A10; 47G20; Primary 53C38; Nonlocal perimeter; Calibration; Secondary 49Q10; Nonlocal minimal surfaces; 49Q20; Viscosity solutions
• ISSN: 0373-3114; 1618-1891
• DOI: 10.1007/s10231-020-00952-z

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 Ω ⊂ R n , we built a calibration for the nonlocal perimeter inside Ω ⊂ R n . The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in Ω 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.