Global heat kernel estimates for fractional Laplacians in unbounded open sets
Global heat kernel estimates for fractional Laplacians in unbounded open sets
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DOI：
10.1007/s00440-009-0256-0
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发表时间：
2009-06
影响因子：
2
通讯作者：
Zhen-Qing Chen;Joshua M. Tokle
中科院分区：
文献类型：
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作者：
Zhen-Qing Chen;Joshua M. Tokle
本文在两类无界C1，1开集（半空间类开集和外开集）中得到了对称α-稳定过程（或等价地，外条件为零的分数Laplacian）的整体尖锐热核估计.这些开放集可以断开。我们特别关注的是对于所有t> 0和.我们的方法是基于这样的想法，即对于xandyinD远离边界并且t足够大，我们可以将pD（t，x，y）与一个很好理解的开集中的热核进行比较：半空间或;而对于一般情况，我们可以通过将xandy推到远离边界的内部来将它们减少到上述情况。因此，在这两种类型的开集上，得到了Dirichlet分数拉普拉斯算子的精确绿色函数估计。对于截尾稳定过程（或等价地，对于区域分数Laplacian），也得到了外开集上的全局锐热核估计和绿色函数估计.
In this paper, we derive global sharp heat kernel estimates for symmetricα-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unboundedC1,1open sets in: half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates forpD(t,x,y) for allt> 0 and. Our approach is based on the idea that forxandyinDfar from the boundary andtsufficiently large, we can comparepD(t,x,y) to the heat kernel in a well understood open set: either a half-space or; while for the general case we can reduce them to the above case by pushingxandyinside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets.