Laplacian截断、流形上的多孔和快速扩散及其他应用
Laplacian cut-offs, porous and fast diffusion on manifolds and other applications
摘要：
我们在流形上构造了具有控制梯度和Laplacian的穷竭函数和截断函数，该流形的里奇曲率由一个（可能是无界的）关于到固定参考点的距离的非正函数从下有界，而不需要对拓扑和内射半径作任何假设.一路上，我们证明了一个推广的李-丘梯度估计，这是独立的利益。然后，我们将我们的截止研究的快速和多孔介质扩散，$L^q$-梯度和自伴性的Schroedinger型运营商的性质。
英文摘要：
We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove a generalization of the Li-Yau gradient estimate which is of independent interest. We then apply our cut-offs to the study of the fast and porous media diffusion, of $L^q$-properties of the gradient and of the self-adjointness of Schroedinger-type operators.
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