08-7 Yannick Sire, Enrico Valdinoci

{ Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result (292K, pdf) Jan 10, 08

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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.

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