Isoperimetric sets in spaces with lower bounds on the Ricci curvature. (July 2022)

• Record Type: Journal Article
• Authors: Antonelli, Gioacchino; Pasqualetto, Enrico; Pozzetta, Marco
• Is Part Of: Nonlinear analysis. Volume 220(2022)
• Publication Date: 2022-07
• Subjects: primary 49Q20 53C23 -- secondary 26B30 26A45 49J40; Regularity theory of volume constrained minimizers -- Isoperimetric sets -- RCD spaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112839
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in RCD spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an RCD ( K, N ) space ( X, d, H N ), with K ∈ R, N ≥ 2, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with ( N − 1 ) -Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in RCD ( K, N ) spaces ( X, d, m ) and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters.