Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result

TL;DR

This paper investigates symmetry properties of solutions to nonlocal fractional Laplacian equations using geometric inequalities, establishing new symmetry results and extending De Giorgi's conjecture to boundary reaction problems.

Contribution

It introduces a geometric Poincaré-type inequality for stable solutions of boundary reaction equations related to fractional Laplacians, leading to novel symmetry results.

Findings

Established a geometric inequality for stable solutions.

Proved symmetry results extending De Giorgi's conjecture.

Analyzed boundary reaction equations with fractional Laplacian connections.

Abstract

We deal with symmetry properties for solutions of nonlocal equations of the type $(-\Delta)^s v= f(v)\qquad {in $\R^n$,}$ where $s \in (0,1)$ and the operator $(-\Delta)^s$ is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation ${-div (x^\a \nabla u)=0 \qquad {on $\R^n\times(0,+\infty)$} -x^\a u_x = f(u) \qquad {on $\R^n\times\{0\}$} $ where $\a \in (-1,1)$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\Gamma_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-\Delta)^{\frac{1-\a}{2}}$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\Gamma_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-\Delta)^{\frac{1-\a}{2}}$. More generally, we study the so-called boundary reaction equations given by ${-div (\mu(x) \nabla u)+g(x,u)=0 {on $\R^n\times(0,+\infty)$} - \mu(x) u_x = f(u) {on $\R^n\times{0}$}$ under some natural assumptions on the diffusion coefficient $\mu$ and on the nonlinearities $f$ and $g$. We prove a geometric formula of Poincar\'e-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.