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Fractional De Giorgi conjecture in dimension 2 via complex-plane methods

Serena Dipierro, João Gonçalves da Silva, Giorgio Poggesi, Enrico Valdinoci

TL;DR

The paper provides a novel proof of the fractional De Giorgi conjecture in dimension two for the full range $s\in(0,1)$ by fusing Farina's complex-variable method with the $s$-harmonic extension of the fractional Laplacian. It also establishes a representation formula for finite-energy weak solutions of a broad class of weighted elliptic equations in the half-space with Neumann boundary conditions, linking extended-space solutions to nonlocal traces. Central to the approach is a complex-variable formulation and Fourier analysis that yield one-dimensional symmetry of extended solutions and, consequently, one-dimensional profiles for the nonlocal trace. The results deepen the connection between local extension problems and nonlocal equations, with potential applications to weighted elliptic problems and nonlocal phase-transition models.

Abstract

We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in $\mathbb{R}^2$ for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the $s$-harmonic extension of the fractional Laplacian in the half-space $\mathbb{R}^{3}_+$ introduced by L. Caffarelli and L. Silvestre in 2007. We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space $\mathbb{R}^{n+1}_+$ under Neumann boundary conditions. This generalizes the $s$-harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace.

Fractional De Giorgi conjecture in dimension 2 via complex-plane methods

TL;DR

The paper provides a novel proof of the fractional De Giorgi conjecture in dimension two for the full range by fusing Farina's complex-variable method with the -harmonic extension of the fractional Laplacian. It also establishes a representation formula for finite-energy weak solutions of a broad class of weighted elliptic equations in the half-space with Neumann boundary conditions, linking extended-space solutions to nonlocal traces. Central to the approach is a complex-variable formulation and Fourier analysis that yield one-dimensional symmetry of extended solutions and, consequently, one-dimensional profiles for the nonlocal trace. The results deepen the connection between local extension problems and nonlocal equations, with potential applications to weighted elliptic problems and nonlocal phase-transition models.

Abstract

We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the -harmonic extension of the fractional Laplacian in the half-space introduced by L. Caffarelli and L. Silvestre in 2007. We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space under Neumann boundary conditions. This generalizes the -harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace.

Paper Structure

This paper contains 8 sections, 18 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Main Theorem']}
  3. Fractional Version of the De Giorgi Conjecture in $\mathbb{R}^2$ and proof of Theorem \ref{['Theorem 1']}
  4. Representation Formula for a Class of Weighted Equations in $\mathbb{R}^{n+1}_+$
  5. Functional spaces, notation and results
  6. Towards the proof of Theorem \ref{['Convolution enters the chat']}
  7. Proofs of Theorem \ref{['Convolution enters the chat']} and Corollary \ref{['Rigidity result']}
  8. Proof of some technical lemmata

Key Result

Theorem 1.2

Let $f \in C^1(\mathbb{R})$ and $u\in C^2(\mathbb{R}^2)$ be a solution of Assume that, for all $x\in \mathbb{R}^2$ and $t>0$, and Then, $u$ is one-dimensional, i.e., there exist $u_0:\mathbb{R}\to \mathbb{R}$ and $\omega \in\mathbb{S}^1$ such that $u(x) = u_0(\omega\cdot x)$.

Theorems & Definitions (38)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['Main Theorem']}
  • Theorem 3.1
  • Lemma 3.2
  • ...and 28 more