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

Gabriele Ferla

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--Cabré 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.

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

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 . 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--Cabré 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.

Paper Structure

This paper contains 5 sections, 11 theorems, 55 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['n-7derivatives']} and some applications
  3. Proof of Theorem \ref{['n-2']}
  4. Possible developments
  5. Further observations

Key Result

Theorem 1.3

If $\varphi: \mathbb{R}^n \to \mathbb{R}$ is a $C^2$ function with $\partial_i\varphi >k$ for some $k \in \mathbb{R}$ and for all $i \in [8,\ldots, n]$, and if $\text{graph}(\varphi)$ is a minimal graph, then $\varphi$ is affine.

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2: General definition with more regularity
  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 1.3
  • Theorem 1.6
  • Proposition 1.4
  • ...and 16 more