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.