Long-range phase coexistence models: recent progress on the fractional Allen-Cahn equation

TL;DR

This paper reviews recent advances in the mathematical analysis of the fractional Allen-Cahn equation, focusing on convergence, energy estimates, and symmetry properties of solutions.

Contribution

It provides a comprehensive overview of recent progress on the fractional Allen-Cahn equation, emphasizing $$-convergence, energy estimates, and symmetry results.

Findings

$$-convergence results established

Energy and density estimates derived

Rigidity and symmetry properties analyzed

Abstract

In this set of notes, we present some recent developments on the fractional Allen-Cahn equation $$ (-\Delta)^s u = u-u^3,$$ with special attention to $\Gamma$-convergence results, energy and density estimates, convergence of level sets, Hamiltonian estimates, rigidity and symmetry results.