## Abstract

We deal with symmetry properties for solutions of nonlocal equations of the type(- Δ)^{s} v = f (v) in R^{n}, where s ∈ (0, 1) and the operator (- Δ)^{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^{α} ∇ u) = 0, on R^{n} × (0, + ∞),; - x^{α} u_{x} = f (u), on R^{n} × {0},) where α ∈ (- 1, 1), y ∈ R^{n}, x ∈ (0, + ∞) and u = u (y, x). This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator Γ_{α} : u |_{∂ R+n + 1} {mapping} - x^{α} u_{x} |_{∂ R+n + 1} is (- Δ)^{frac(1 - α, 2)}. More generally, we study the so-called boundary reaction equations given by{(- div (μ (x) ∇ u) + g (x, u) = 0, on R^{n} × (0, + ∞),; - μ (x) u_{x} = f (u), on R^{n} × {0}) under some natural assumptions on the diffusion coefficient μ and on the nonlinearities f and g. We prove a geometric formula of Poincaré-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.

Original language | English |
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Pages (from-to) | 1842-1864 |

Number of pages | 23 |

Journal | Journal of Functional Analysis |

Volume | 256 |

Issue number | 6 |

DOIs | |

Publication status | Published - 15 Mar 2009 |

Externally published | Yes |