The fundamental solution of the Keldysh typeoperator

作     者：CHEN ShuXing School of Mathematical Sciences, Fudan University, Shanghai 200433, China 

作者机构：School of Mathematical Sciences Fudan University Shanghai China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2009年第52卷第9期

页      面：1829-1843页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by the National Basic Research Program of China (Grant No.2006CB805902) National Natural Science Foundation of China (Grant No.10531020) the Research Foundation for Doctor Programme (Grant No.20050246001) 

主　　题：mixed type equation Keldysh operator fundamental solution finite part of divergent integral 35M10 35A08 

摘      要：In this paper we discuss the fundamental solution of the Keldysh type operator $ L_\alpha u \triangleq \frac{{\partial ^2 u}} {{\partial x^2 }} + y\frac{{\partial ^2 u}} {{\partial y^2 }} + \alpha \frac{{\partial u}} {{\partial y}} $ , which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator with $ \alpha - \frac{1} {2} $ is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with $ \alpha \frac{1} {2} $ has to be defined by using the finite part of divergent integrals in the theory of distributions.