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arXiv 0801.2355math.APmath.FA
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分数拉普拉斯相变和边界反应：一个几何不等式和一个对称性结果

Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result

Yannick Sire, Enrico Valdinoci

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摘要：

我们处理类型为$(-Δ)^s v= f(v)\qquad {in $ \R^n $,}$的非局部方程解的对称性，其中$s \in (0,1)$和运算符$(-Δ)^s$是所谓的分数拉普拉斯算子。通过对退化椭圆型方程${-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\} $} $的分析，研究了该非局部方程，其中$\a \in (-1,1)$ .这个方程与分数拉普拉斯算子有关，因为狄利克雷-诺依曼算子$Γ_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $是$(-Δ)^{\frac{1-\a}{2}}$。这个方程与分数拉普拉斯算子有关，因为狄利克雷-诺依曼算子$Γ_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $是$(-Δ)^{\frac{1-\a}{2}}$。更一般地，我们研究了所谓的边界反应方程给出的${-div (μ(x) \nabla u)+g(x,u)=0 {on $ \R^n \times（0，+ \infty）$} - μ(x) u_x = f(u) {on $ \R^n \times{0} $}$在一些自然的假设下的扩散系数$μ$和非线性$f$和$g$。我们证明了一个几何公式的庞加莱型稳定的解决方案，从中我们得到一个对称性结果的德Giorgi猜想的精神。

英文摘要：

We deal with symmetry properties for solutions of nonlocal equations of the type $(-Δ)^s v= f(v)\qquad {in $\R^n$,}$ where $s \in (0,1)$ 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^\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 $Γ_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-Δ)^{\frac{1-\a}{2}}$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $Γ_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-Δ)^{\frac{1-\a}{2}}$. More generally, we study the so-called boundary reaction equations given by ${-div (μ(x) \nabla u)+g(x,u)=0 {on $\R^n\times(0,+\infty)$} - μ(x) u_x = f(u) {on $\R^n\times{0}$}$ 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.

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