论文标题
对称性和对称性破坏分数Caffarelli-Kohn-Nirenberg不平等
Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality
论文作者
论文摘要
在本文中，我们将考虑分数Caffarelli-kohn-nirenberg不等式\ begin {equation*}λ\ left（\ int _ {\ m athbb r^n} \ frac {| u（x） {p}}}} \，dx \ right）^{\ frac {2} {p}}} \ leq \ int _ {\ mathbb r^n} \ int \ int \ int _ {\ mathbb r^n} \ frac {（u（x）-u（y））^2} {| x-y |^{n+2γ} | x | x |^{α} | y | y |^{α}}} \，dy \，dx \，dx \ end end end eent {earkation*} $α，β\ in \ Mathbb r $满足\ begin {等式*}α\ leqβ\ leqβ\ leqα+γ，\-2γ<α<α<\ frac {n -2γ} {2} {2} {2} {2} {2}，\ end End {earkation*} and equent $ p $ w begenation w begin w begin} equestion* p = \ frac {2n} {n-2γ+2（β-α）}，\ end {方程*}，使得不等式在缩放下是不变的。我们首先研究了极端解决方案的存在和不存在。我们的下一个目标是在最小化器的对称性和对称破坏区域上显示一些结果；这些表明存在将两个区域分开的Felli -Schneider型曲线的存在，但令人惊讶的是，我们发现一种新颖的行为是$α\至-2γ$。与经典案例一样，证明中的主要思想是重新重新制定圆柱变量中的咖啡馆 - 科恩·尼伦贝格的不平等。然后，为了找到径向对称解，我们需要解决非本地颂歌。 对于这个方程式，我们还获得了径向对称类别中最小化器的唯一性。确实，我们表明，Frank-Lenzmann（Acta'13）的独特延续论证可以应用于具有良好频谱属性的更普通运营商。此外，我们还提供了一个全新的非分类证明，可适用于所有关键点。它基于常数方法的变化和Ao-Chan-Delatorre-Fontelos-González-Wei的非本地Wronskian（Duke'19）。
In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality \begin{equation*} Λ \left(\int_{\mathbb R^n}\frac{|u(x)|^{p}}{|x|^{β {p}}}\,dx\right)^{\frac{2}{p}}\leq \int_{\mathbb R^n}\int_{\mathbb R^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2γ}|x|^{α}|y|^{α}}\,dy\,dx \end{equation*} where $γ\in(0,1)$, $n\geq 2$, and $α,β\in\mathbb R$ satisfy \begin{equation*} α\leq β\leq α+γ, \ -2γ<α<\frac{n-2γ}{2}, \end{equation*} and the exponent $p$ is chosen to be \begin{equation*} p=\frac{2n}{n-2γ+2(β-α)}, \end{equation*} such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as $α\to -2γ$. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE. For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of Frank-Lenzmann (Acta'13) can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non-local Wronskian of Ao-Chan-DelaTorre-Fontelos-González-Wei (Duke'19).
