On critical points of the relative fractional perimeter

2018/02/05 by Andrea Malchiodi, Malchiodi, Andrea, Matteo Novaga +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1802.01510

openalex publication_date 2018/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are \em sufficiently close to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the existence of a minimizer under fixed volume constraint, showing some of its properties such as smoothness and symmetry, being a graph in the x_N-direction, and characterizing its intersection with the hyperplane \x_N=0\.

Citations

• Hitchhikerʼs guide to the fractional Sobolev spaces
• On category, in the sense of Lusternik-Schnirelmann

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