Some perspectives on (non)local phase transitions and minimal surfaces

2022/07/11 by Serena Dipierro, Enrico Valdinoci, Dipierro, Serena +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2207.04783

openalex publication_date 2022/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present here some classical and modern results about phase transitions and minimal surfaces, which are quite intertwined topics. We start from scratch, revisiting the theory of phase transitions as put forth by Lev Landau. Then, we relate the short-range phase transitions to the classical minimal surfaces, whose basic regularity theory is presented, also in connection with a celebrated conjecture by Ennio De Giorgi. With this, we explore the recently developed subject of long-range phase transitions and relate its genuinely nonlocal regime to the analysis of fractional minimal surfaces.

Cited by

• Nonlocal free boundary minimal surfaces
• Heteroclinic connections for fractional Allen-Cahn equations with degenerate potentials

Related

• A comparison between the nonlocal and the classical worlds: minimal surfaces, phase transitions, and geometric flows
• Fractional multi-phase transitions and nonlocal minimal partitions
• Minimality of level sets in phase transitions
• Min-max for phase transitions and the existence of embedded minimal\n hypersurfaces
• Fractional generalization of the Ginzburg–Landau equation: an unconventional approach to critical phenomena in complex media