Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Stable Solutions to the Free Boundary Allen--Cahn and Bernoulli Problems in 3D are One-Dimensional

Hardy Chan, Xavier Fernández-Real, Alessio Figalli, Joaquim Serra

TL;DR

The paper proves that in three dimensions, global stable solutions to the free boundary Allen–Cahn equation with a step potential are one-dimensional, by reducing the problem to a Bernoulli free boundary setting and establishing universal curvature bounds. The authors develop a novel ball-tree decomposition of the positivity set, identify neck centers and radii, and connect neck radii to a symmetric excess via a Jerison–Savin-type test function, enabling dichotomous excess decay. A blow-down analysis shows that non-flat global Bernoulli solutions converge to a vee at infinity, which, together with stability and linearization, yields a contradiction unless the solution is planar. As corollaries, the De Giorgi conjecture is verified in dimension 3 for the free boundary model and, consequently, the monotone De Giorgi conjecture holds in dimension 4 for corresponding solutions, with universal curvature bounds for local stable Bernoulli solutions. The results bridge free boundary theory and minimal-surface techniques, offering a robust framework for extending one-dimensional rigidity from the Bernoulli problem to the free boundary Allen–Cahn equation.

Abstract

A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in \(\mathbb{R}^{n+1}\) is one-dimensional if \(n \leq 7\). A stronger version of the conjecture, also widely studied and often called ``the stable De Giorgi conjecture'', proposes that every stable solution in \(\mathbb{R}^n\) must be one-dimensional for \(n \leq 7\). To this date, both conjectures remain open for \(3 \leq n \leq 7\). An elegant variant of this problem, advocated by Caffarelli, Córdoba, and Jerison since the 1990s, considers a free boundary version of the Allen--Cahn equation. This variant features a step-like double-well potential, leading to multiple free boundaries. Locally, near each free boundary, the solution satisfies the Bernoulli free boundary problem. However, the interaction of the free boundaries causes the global behavior of the solution to resemble that of the Allen--Cahn equation. In this paper, we establish the validity of the stable De Giorgi conjecture in dimension 3 for the free boundary Allen--Cahn equation and, as a corollary, we prove the corresponding De Giorgi conjecture for monotone solutions in dimension 4. To obtain these results, a key aspect of our work is to address a classical open problem in free boundary theory of independent interest: the classification of global stable solutions to the one-phase Bernoulli problem in three dimensions. This result, which is the core of our paper, implies universal curvature estimates for local stable solutions to Bernoulli, and serves as a foundation for adapting some classical ideas from minimal surface theory -- after significant refinements -- to the free boundary Allen--Cahn equation.

Global Stable Solutions to the Free Boundary Allen--Cahn and Bernoulli Problems in 3D are One-Dimensional

TL;DR

The paper proves that in three dimensions, global stable solutions to the free boundary Allen–Cahn equation with a step potential are one-dimensional, by reducing the problem to a Bernoulli free boundary setting and establishing universal curvature bounds. The authors develop a novel ball-tree decomposition of the positivity set, identify neck centers and radii, and connect neck radii to a symmetric excess via a Jerison–Savin-type test function, enabling dichotomous excess decay. A blow-down analysis shows that non-flat global Bernoulli solutions converge to a vee at infinity, which, together with stability and linearization, yields a contradiction unless the solution is planar. As corollaries, the De Giorgi conjecture is verified in dimension 3 for the free boundary model and, consequently, the monotone De Giorgi conjecture holds in dimension 4 for corresponding solutions, with universal curvature bounds for local stable Bernoulli solutions. The results bridge free boundary theory and minimal-surface techniques, offering a robust framework for extending one-dimensional rigidity from the Bernoulli problem to the free boundary Allen–Cahn equation.

Abstract

A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen--Cahn equation in is one-dimensional if . A stronger version of the conjecture, also widely studied and often called ``the stable De Giorgi conjecture'', proposes that every stable solution in must be one-dimensional for . To this date, both conjectures remain open for . An elegant variant of this problem, advocated by Caffarelli, Córdoba, and Jerison since the 1990s, considers a free boundary version of the Allen--Cahn equation. This variant features a step-like double-well potential, leading to multiple free boundaries. Locally, near each free boundary, the solution satisfies the Bernoulli free boundary problem. However, the interaction of the free boundaries causes the global behavior of the solution to resemble that of the Allen--Cahn equation. In this paper, we establish the validity of the stable De Giorgi conjecture in dimension 3 for the free boundary Allen--Cahn equation and, as a corollary, we prove the corresponding De Giorgi conjecture for monotone solutions in dimension 4. To obtain these results, a key aspect of our work is to address a classical open problem in free boundary theory of independent interest: the classification of global stable solutions to the one-phase Bernoulli problem in three dimensions. This result, which is the core of our paper, implies universal curvature estimates for local stable solutions to Bernoulli, and serves as a foundation for adapting some classical ideas from minimal surface theory -- after significant refinements -- to the free boundary Allen--Cahn equation.

Paper Structure

This paper contains 57 sections, 93 theorems, 626 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The variational model of phase transitions
  3. De Giorgi conjecture and its stable version
  4. The connection to minimal surfaces
  5. Recent progress and challenges in stable phase transitions
  6. The Allen--Cahn model with a step potential
  7. Microscopic necks: a new "enemy."
  8. The one-phase Bernoulli problem
  9. Contributions of the paper
  10. Structure of the paper
  11. Overview of the Proofs
  12. The Bernoulli problem: Sections \ref{['sec:Bernoulli']}--\ref{['sec:closing']}
  13. The objects and their properties
  14. Sketch of the global contradiction argument
  15. The free boundary Allen--Cahn: Section \ref{['sec:FB_AC']}
  16. ...and 42 more sections

Key Result

Theorem 1.1

Let $u: \mathbb{R}^3 \to [-1,1]$ be a classical stable critical point of $\mathcal{J}^0_1$ (i.e., a global classical stable solution of freebdryAC, see defi:solutionsAC). Then $D^2u \equiv 0$ in $\{|u|<1\}$ and $u$ is one-dimensional.

Figures (1)

  • Figure 1: Illustration of branching structure in ball tree: \ref{['prop:geomtree']} and \ref{['defiregball']}. From left to right: a neck-type terminal ball, a regular terminal ball, and a branching ball.

Theorems & Definitions (202)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark 1.8
  • Definition 3.1
  • Lemma 3.2
  • ...and 192 more