刘维尔定理中最优幂的一个简单证明

Pub Date : 2020-03-09 DOI:10.5565/publmat6622212

S. Villegas

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引用次数: 1

摘要

考虑方程div$(\varphi^2 \nabla \sigma)=0$ 在 $\mathbb{R}^N,$ 在哪里 $\varphi>0$．众所周知，如果存在的话 $C>0$ 这样 $\int_{B_R}(\varphi \sigma)^2 dx\leq CR^2$ 对于每一个 $R\geq 1$ 然后 $\sigma$ 必然是常数。在本文中，我们证明了如果进行替换，这个结果是不成立的 $R^2$ 通过 $R^k$ 为了 $k>2$ 在任何维度上 $N$．这个问题与德·乔治的一个猜想有关。

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A simple proof of the optimal power in Liouville theorems

Consider the equation div$(\varphi^2 \nabla \sigma)=0$ in $\mathbb{R}^N,$ where $\varphi>0$. It is well-known that if there exists $C>0$ such that $\int_{B_R}(\varphi \sigma)^2 dx\leq CR^2$ for every $R\geq 1$ then $\sigma$ is necessarily constant. In this paper we prove that this result is not true if we replace $R^2$ by $R^k$ for $k>2$ in any dimension $N$. This question is related to a conjecture by De Giorgi.

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