二阶混合型方程的边值问题

Boundary Value Problems for Mixed Equations of Second Order

在文献[1-5]中,作者提出和讨论了具有抛物退化线段的二阶混合型方程的Tricomi问题,它对于空气动力学有重要应用。本文中,我们先给出广义混合型(Lavrent’ev-Bitsadze)方程边值问题解的表达式,该边值问题包括Tricomi问题作为特殊情况,然后证明上述问题解的唯一性。最后我们证明广义二阶混合型(Lavrent’ev-Bitsadze)方程边值问题的可解性,从而得到上述问题解的先验估计。上述问题是一个类似于由J.M.Rassias在最近提出的一个未解解决的问题。

In [1--5], the authors posed and discussed the Tricomi problem of second order e- quations of mixed type with parabolic degenerate line segments, which possesses the impor- tant applications to gas dynamics. In this article, we first give the representation of solutions for boundary value problems of generalized mixed （Lavrentrev-Bitsadze） equation, the bound- ary value problems include the Tricomi problem as a special case, afterwards prove the u- niqueness of solutions of the above problem. Finally we prove the solvability of boundary val- ue problems for generalized mixed （Lavrent＇ev-Bitsadze） equations of second order, and then obtain a priori estimates of solutions of the above problems. The above problem is an open problem similar to that proposed by J. M. Rassias recently.