Harnack Inequalities and Heat-kernel Estimates for Degenerate Diffusion Operators Arising in Population Biology
Harnack Inequalities and Heat-kernel Estimates for Degenerate Diffusion Operators Arising in Population Biology
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群体生物学中简并扩散算子的 Harnack 不等式和热核估计
DOI：
10.1093/amrx/abw002
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
R. Mazzeo
中科院分区：
文献类型：
--
作者：
C. Epstein;R. Mazzeo
本文继续分析了在文献[2，3]中提出的定义在角流形上的一类退化椭圆算子，这类算子是在种群生物学中出现的。使用由J.莫泽开创的技术，并由L. Saloff-Coste，Grigoryan，and Sturm，我们证明了由这些算子的一个子类定义的抛物问题的弱解满足Harnack不等式，该算子的子类包括那些可以由Dirichlet形式定义并且具有非零横向向量场的算子。这使我们能够得出结论，这些方程的解决方案属于，为积极的时间，自然各向异性的保持器空间，也导致了上限，在某些情况下，这些运营商的热核的下限。这些结果意味着这些算子在作用于$C^0$或$L^2 $时有紧预解式。证明依赖于一个规模不变的庞加莱不等式，我们建立了一个大类的加权狄利克雷形式，以及估计处理某些温和的奇异扰动项。我们考虑的权既不是Ahlfors正则的，也不是一般地属于Muckenhaupt类$A_2。
This paper continues the analysis, started in [2, 3], of a class of degenerate elliptic operators defined on manifolds with corners, which arise in Population Biology. Using techniques pioneered by J. Moser, and extended and refined by L. Saloff-Coste, Grigoryan, and Sturm, we show that weak solutions to the parabolic problem defined by a sub-class of these operators, which consists of those that can be defined by Dirichlet forms and have non-vanishing transverse vector field, satisfy a Harnack inequality. This allows us to conclude that the solutions to these equations belong, for positive times, to the natural anisotropic Holder spaces, and also leads to upper and, in some cases, lower bounds for the heat kernels of these operators. These results imply that these operators have a compact resolvent when acting on $C^0$ or $L^2.$ The proof relies upon a scale invariant Poincare inequality that we establish for a large class of weighted Dirichlet forms, as well as estimates to handle certain mildly singular perturbation terms. The weights that we consider are neither Ahlfors regular, nor do they generally belong to the Muckenhaupt class $A_2.$