论文标题
最佳Sobolev不平等的刚性定理
Rigidity theorems for best Sobolev inequalities
论文作者
论文摘要
对于$ n \ geq 2 $，$ p \ in（1，n）$，在开放式套装$ po $ -sobolev不平等的“最佳$ p $ - sobolev”中，$ω\ subset \ subset \ mathbb {r}^n $与一个家庭$φ_Ω$具有关键体积和跟踪约束的各种问题的$φ_Ω$。当$ω$有限时，我们证明：（i）对于每个$ n $和$ p $，存在最多具有一个边界浓度点的广义极小化器，以及：（ii）对于$ n> 2 \，p $，存在（经典）最小化器。然后，我们为比较定理建立了刚性的结果“球是第一位命名作家和维拉尼的最佳Sobolev不平等现象”，从而为[MV05]提出的问题提供了第一个肯定的答案。
For $n\geq 2$, $p\in(1,n)$, the "best $p$-Sobolev inequality" on an open set $Ω\subset\mathbb{R}^n$ is identified with a family $Φ_Ω$ of variational problems with critical volume and trace constraints. When $Ω$ is bounded we prove: (i) for every $n$ and $p$, the existence of generalized minimizers that have at most one boundary concentration point, and: (ii) for $n> 2\,p$, the existence of (classical) minimizers. We then establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities" by the first named author and Villani, thus giving the first affirmative answers to a question raised in [MV05].
