论文标题
Liouville定理的最佳力量的简单证明
A simple proof of the optimal power in Liouville theorems
论文作者
论文摘要
考虑方程式div $（φ^2 \nablaσ）= 0 $ in $ \ mathbb {r}^n，$ $φ> 0 $。众所周知，如果存在$ c> 0 $，则$ \ int_ {b_r}（φσ）^2 dx \ leq cr^2 $对于每个$ r \ geq 1 $ 1 $，那么$σ$一定是恒定的。在本文中，我们证明，如果我们将$ r^2 $替换为$ r^k $，$ k> 2 $在任何尺寸$ n $中，则该结果是不正确的。这个问题与de Giorgi的猜想有关。
Consider the equation div$(φ^2 \nabla σ)=0$ in $\mathbb{R}^N,$ where $φ>0$. It is well-known that if there exists $C>0$ such that $\int_{B_R}(φσ)^2 dx\leq CR^2$ for every $R\geq 1$ then $σ$ 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.
