论文标题
夏普liouville定理
Sharp 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 $。 Berestycki，Caffarelli和Nirenberg证明，如果存在$ c> 0 $，则每一个$ \ int_ {b_r}（φσ）^2 \ leq cr^2 $对于每一个$ r \ geq 1 $，那么$σ$就必须稳定。在本文中，我们在c（[1，\ infty））$ $ 0 <ψ\ $ 0 <ψ\上提供了必要的条件，如果我们将$ r^2 $替换为任何尺寸$ n $中的$ r^2 $，则将其结果为此。对于大$ r> 1 $和$ψ'> 0 $的$ψ$的凸度，此条件相当于$ \ displayStyle {\ int_1^\ int_1^\ infty \ frac {1} {ψ'} = \ infty} $。
Consider the equation div$(φ^2 \nabla σ)=0$ in $\mathbb{R}^N,$ where $φ>0$. Berestycki, Caffarelli and Nirenberg proved that if there exists $C>0$ such that $\int_{B_R}(φσ)^2 \leq CR^2$ for every $R\geq 1$ then $σ$ is necessarily constant. In this paper we provide necessary and sufficient conditions on $0<Ψ\in C([1,\infty))$ for which this result remains true if we replace $R^2$ with $Ψ(R)$ in any dimension $N$. In the case of the convexity of $Ψ$ for large $R>1$ and $Ψ'>0$, this condition is equivalent to $\displaystyle{\int_1^\infty\frac{1}{Ψ'}=\infty}$.
