Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fractional Laplacian phase transitions and boundary reactions: a
  geometric inequality and a symmetry result   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.

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The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol