Non-Local Isoperimetric Problems

2014/06/29 by Agnese Di Castro, Berardo Ruffini, Di Castro, Agnese +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1406.7545

openalex publication_date 2014/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We characterize the volume-constrained minimizers of a nonlocal free energy\ngiven by the difference of the t-perimeter and the s-perimeter, with s\nsmaller than t. Exploiting the quantitative fractional isoperimetric\ninequality, we show that balls are the unique minimizers if the volume is\nsufficiently small, depending on t-s, while the existence vs. nonexistence of\nminimizers for large volumes remains open. We also consider the corresponding\nisoperimetric problem and prove existence and regularity of minimizers for all\ns, ,t. When s=0 this problem reduces to the fractional isoperimetric\nproblem, for which it is well known that balls are the only minimizers.\n

Citations

• Regularity for non-local almost minimal boundaries and applications

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