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

TL;DR

This paper explores the theoretical connections between phase transitions and minimal surfaces, covering classical and modern results, including Landau's theory, De Giorgi's conjecture, and fractional minimal surfaces.

Contribution

It provides a comprehensive overview linking local and nonlocal phase transitions with minimal surface theory, highlighting recent developments and open problems.

Findings

Revisits Landau's phase transition theory

Connects short-range phase transitions to classical minimal surfaces

Relates long-range phase transitions to fractional minimal surfaces

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.