论文标题
梯度估计值在RICCI范围
Gradient estimates under integral Ricci bounds
论文作者
论文摘要
在本文中，我们研究了Riemannian歧管上$ΔU= f $的解决方案的全球规则性估计。在RICCI张量的积分（较低）范围下，我们证明了$ l^p $ - 形式$ ||的有效性估计值\ nabla u || _ {l^p} \ le c（|| u || _ {l^p} + ||ΔU|| _ {l^p}）$。我们还构建了一个反例，证明了RICCI曲率上先前已知的常数下限在侧面意义上是最佳的。还研究了$ l^p $ - 级别估计与Sobolev空间不同概念之间的关系。
In this paper we study $W^{1,p}$ global regularity estimates for solutions of $Δu = f$ on Riemannian manifolds. Under integral (lower) bounds on the Ricci tensor we prove the validity of $L^p$-gradient estimates of the form $|| \nabla u ||_{L^p} \le C (|| u ||_{L^p} + || Δu||_{L^p})$. We also construct a counterexample which proves that the previously known constant lower bounds on the Ricci curvature are optimal in the pointwise sense. The relation between $L^p$-gradient estimates and different notions of Sobolev spaces is also investigated.
