Nonlocal phase transitions: rigidity results and anisotropic geometry

TL;DR

This paper establishes rigidity and symmetry results for nonlocal phase transition equations, including anisotropic cases, advancing understanding of solutions' geometric properties and contributing to the De Giorgi conjecture.

Contribution

It improves flatness theorems and proves one-dimensional symmetry for monotone and minimal solutions in nonlocal phase transitions, including anisotropic operators.

Findings

Enhanced flatness theorem for nonlocal equations

One-dimensional symmetry for monotone solutions

Rigidity results for anisotropic nonlocal operators

Abstract

We provide a series of rigidity results for a nonlocal phase transition equation. The prototype equation that we consider is of the form $$ (-\Delta)^{s/2} u=u-u^3,$$ with~$s\in(0,1)$. More generally, we can take into account equations like $$ L u = f(u),$$ where $f$ is a bistable nonlinearity and $L$ is an integro-differential operator, possibly of anisotropic type. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in the companion paper.