A Matrix Differential Harnack Estimate for a Class of Ultraparabolic Equations

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• 作者：Hong Huang
• 关键词：Ultraparabolic equation ; Matrix differential Harnack estimate ; Maximum principle ; 35K70
• 刊名：Potential Analysis
• 出版年：2014
• 出版时间：October 2014
• 年：2014
• 卷：41
• 期：3
• 页码：771-782
• 全文大小：328 KB
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• 作者单位：Hong Huang (1)
  
  1. School of Mathematical Sciences, Key Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, People’s Republic of China
  
• ISSN：1572-929X

文摘

Let u be a positive solution of the ultraparabolic equation $$\partial _{t} u=\sum\limits_{i=1}^{n} \partial _{x_{i}}^{2} u+\sum\limits_{i=1}^{k} x_{i}\partial _{x_{n+i}}u \hspace {8mm} \text {on} \hspace {4mm} \mathbb {R}^{n+k}\times (0,T),$$ where 1 ?k ?n and 0 T ?+ ?/em>. Assume that u and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of (0, T). Then the difference H(log u) ?H (log f) of the Hessian matrices of log u and of log f (both w.r.t. the space variables) is non-negatively definite, where f is the fundamental solution of the above equation with pole at the origin (0, 0). The estimate in the case n = k = 1 is due to Hamilton. As a corollary we get that \(\Delta l+\frac {n+3k}{2t}+\frac {6k}{t^{3}}\geq 0\) , where l = log u, and \(\Delta =\sum _{i=1}^{n+k} \partial _{x_{i}}^{2} \) .