Yannick Sire, Enrico Valdinoci
{ 
Fractional Laplacian phase transitions 
and boundary reactions: 
a geometric inequality 
and a symmetry result
(292K, pdf)

ABSTRACT.  We deal with symmetry 
properties for solutions 
of nonlocal equations of the type 
\begin{equation*} 
(-\Delta)^s v= f(v)\qquad 
{\mbox{ in $\R^n$,}} 
\end{equation*} 
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 
$$ 
\left\{ 
\begin{matrix} 
-{\rm div}\, (x^\a \nabla u)=0 \qquad 
{\mbox{ on $\R^n\times(0,+\infty)$}} 
\\ 
-x^\a u_x = f(u) 
\qquad{\mbox{ on $\R^n\times\{0\}$}}\end{matrix} 
\right.$$ 
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}} $. 
More generally, we study the so-called boundary reaction equations 
given by 
 \begin{equation*}\left\{ 
\begin{matrix} 
-{\rm div}\, (\mu(x) \nabla u)+g(x,u)=0 \qquad 
{\mbox{ on $\R^n\times(0,+\infty)$}} 
\\ 
-\mu(x) u_x = f(u) 
\qquad{\mbox{ on $\R^n\times\{0\}$}}\end{matrix} 
\right.\end{equation*} 
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.