Rigidity of minimizers in nonlocal phase transitions II

TL;DR

This paper extends previous results to the critical case s=1/2, classifying bounded solutions with flat level sets for nonlocal equations involving the half-Laplacian and a double well potential.

Contribution

It provides a classification of solutions in the borderline case s=1/2, advancing understanding of nonlocal phase transition models.

Findings

Classified global bounded solutions with flat level sets for s=1/2

Extended previous results to the critical case s=1/2

Analyzed solutions of nonlocal equations with double well potential

Abstract

In this paper we extend the results of \cite{S3} to the borderline case $s = \frac 12$. We obtain the classification of global bounded solutions with asymptotically flat level sets for semilinear nonlocal equations of the type $$\Delta^{\frac 12} u=W'(u) \quad \text{in} \quad R^n,$$ where $W$ is a double well potential.