Rigidity of minimizers in nonlocal phase transitions

TL;DR

This paper classifies global bounded solutions of semilinear nonlocal equations involving fractional Laplacians with order s in (1/2, 1), providing insights into phase transition models with double well potentials.

Contribution

It offers a classification of solutions for a class of nonlocal equations, extending understanding of phase transitions with fractional operators.

Findings

Classification of solutions for nonlocal equations with fractional Laplacian.

Extension of phase transition theory to nonlocal operators.

Results applicable to models with double well potentials.

Abstract

We obtain the classification of certain global bounded solutions for semilinear nonlocal equations of the type $$\triangle^s u=W'(u)$$ in $\mathbb{R}^n$,with $s \in (1/2 ,1),$ where $W$ is a double well potential.