The fundamental solution of the Keldysh typeoperator  

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

出　　处：《Science China Mathematics》2009年第9期1829-1843,共15页中国科学：数学（英文版）

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)

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.In this paper we discuss the fundamental solution of the Keldysh type operator Lαu■ 2u x2 + y 2yu y2 + α u y1 , which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator with α>-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 α<1/2 has to be defined by using the finite part of divergent integrals in the theory of distributions.

关 键 词：mixed type equation Keldysh operator fundamental solution finite part of divergent integral 35M10 35A08 

分 类 号：O177[理学—数学]