2023/08/03 by Serena Dipierro, Ovidiu Savin, Dipierro, Serena +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2308.01697
openalex publication_date 2023/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide. This strict maximum principle is not obvious, since the surfaces may touch at an irregular point, therefore a suitable blow-up analysis must be combined with a bespoke regularity theory to obtain this result. For the classical case, an analogous result was proved by Leon Simon. Our proof also relies on a Harnack Inequality for nonlocal minimal surfaces that has been recently introduced by Xavier Cabré and Matteo Cozzi and which can be seen as a fractional counterpart of a classical result by Enrico Bombieri and Enrico Giusti. In our setting, an additional difficulty comes from the analysis of the corresponding nonlocal integral equation on a hypersurface, which presents a remainder whose sign and fine properties need to be carefully addressed.