Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

TL;DR

This paper proves sharp energy estimates for solutions of a fractional nonlinear equation involving the half-Laplacian, leading to the one-dimensional symmetry of certain solutions in three dimensions, extending classical symmetry results.

Contribution

It establishes optimal energy estimates for solutions of fractional equations and proves their one-dimensional symmetry in three dimensions, confirming a fractional analog of De Giorgi's conjecture.

Findings

Energy estimates are sharp and optimal for 1D solutions.

Global minimizers and monotone solutions in 3D are one-dimensional.

Extends classical symmetry results to fractional Laplacian equations.

Abstract

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)^{1/2} u=f(u)$ in $\re^n$. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in $\re^n$.