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. 本年度学院报告总序号： 233