Harnack and Pointwise Estimates for Degenerate or Singular Parabolic Equations
Harnack and Pointwise Estimates for Degenerate or Singular Parabolic Equations
复制标题
简并或奇异抛物线方程的 Harnack 和逐点估计
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
V. Vespri
中科院分区：
文献类型：
--
作者：
F. Duzgun;S. Mosconi;V. Vespri
在本文中，我们给出了一个历史和技术的概述理论的Harnack不等式的非线性抛物方程的散度形式。我们开始回顾椭圆的情况下，它的一些变种和几何后果。广泛地讨论了线性抛物型的Moser Harnack不等式，以及它与双边核估计和Li-Yau微分Harnack不等式的联系。然后，我们概述了非线性退化/奇异方程理论的最新发展，突出了与二次情况下的差异，并介绍了所谓的内在Harnack不等式。最后，我们给出了Harnack不等式在某些重要情形下的完整证明，以向读者介绍正性方法的扩展。
In this paper we give both a historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developments of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introducing the so-called intrinsic Harnack inequalities. Finally, we provide complete proofs of the Harnack inequalities in some paramount case to introduce the reader to the expansion of positivity method.