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发表时间:2018-10-24 阅读次数:89次
报告题目: Cauchy-Kowalevski and Holmgren type theorems on $\mathbb{R}^{\infty}$
报 告 人:Dr. Jiayang Yu
报告人所在单位:四川大学
报告日期:2018-10-24 星期三
报告时间:11:00-12:00
报告地点:H6311
  
报告摘要:

We will give a brief report of Cauchy-Kowalevski and Holmgren type theorems on $\mathbb{R}^{\infty}$ that we have got. We adapt the definition of analyticity of functions on $\mathbb{R}^{\infty}$ by Hilbert as monomial expansions.  To describe analyticity a family of $\ell^p$ induced topologies on $\mathbb{R}^{\infty}$ and corresponding topologies on $\mathbb{R}^{\infty}$ with infinity point are given. By the method of Friedman and majorants we get two distinct Cauchy-Kowalevski type theorems on $\mathbb{R}^{\infty}$. Based on the Cauchy-Kowalevski type theorem we have established, the tools of abstract Wiener spaces, Malliavin analysis and a divergence theorem for Hilbert space, we obtain a Holmgren type theorem. There are two reasons that the known results for Cauchy-Kowalevski type theorems on Banach spaces can not imply our results. Firstly, $\mathbb{R}^{\infty}$ is not a Banach space under the topology we considered. Secondly, we use partial derivatives and a family of locally Fr\'{e}chet derivatives instead of only one Fr\'{e}chet derivative in the known results. Then the usual method to reduce the order of the equation to get a equivalent system of finite equations does not work.

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