Some remarks on the Allen-Cahn equation in $\mathbb{R}^n$

TL;DR

This paper extends symmetry results for Allen-Cahn equations in high dimensions, identifying conditions under which solutions are one-dimensional, and explores related nonlocal and free boundary problems.

Contribution

It proves a Savin-type theorem in arbitrary dimensions for energy-minimizing solutions with monotonic level sets, and extends results to nonlocal transitions and non-minimizers.

Findings

Proved one-dimensional symmetry under n-7 monotonic directions for energy minimizers.

Extended symmetry results to nonlocal phase transitions.

Established a version of the Ambrosio–Cabré theorem for solutions lacking bounded energy density.

Abstract

In this short note we present new results on a higher-dimensional generalization of De~Giorgi's conjecture for Allen--Cahn type equations, focusing on dimensions $n \ge 9$. Although counterexamples are known in this regime, our goal is to identify assumptions on solutions that still enforce one-dimensional symmetry. We prove an analogue of Savin's theorem in arbitrary dimension: for energy-minimizing solutions whose level sets enjoy n-7 directions of monotonicity, we deduce one-dimensional symmetry. In the same spirit, we extend these ideas to nonlocal phase transitions, and we discuss an application to free boundary problems Finally, we establish a counterpart of the Ambrosio--Cabr\'e theorem for solutions that are not necessarily energy minimizers and may lack bounded energy density, assuming instead n-2 directions of monotonicity everywhere. Overall, this note aims to further strengthen the connection between phase transition models and minimal surface theory.