一般形式的一阶椭圆型偏微分方程组拟线性Riemann-Hilbert问题

A CLASS OF QUASI-LINEAR RIEMANN-HILBERT PROBLEMS FOR SYSTEM OF FIRST ORDER ELLIPTIC EQUATIONS WITH GENERAL FORM

在文［ｌ，２，３］中，Ｅ．Ｗｅｇｅｒｔ和Ｌ．Ｖ．Ｗｏｌｆｅｒｓｄｏｒｆ等人讨论了一类全纯函数的拟线性Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ 问题在 Ｈａｒｄｙ空间中的可解性，在文［４］中，讨论了广义解析函数的拟线性 Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ问题，同样得到该边值问题在Ｈ２类解空间中的可解性、本文在前面研究工作的基础上，对一般形式的一阶椭圆型偏微分方程组拟线性Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ问题作了更深入的讨论，在适当的假设条件下，应用积分算子理论，函数论方法及不动点原理，证明了该边值问题在相应的泛函空间中同样是可解的．

In paper [1.2.3], E. Wegert and L. V. Wolfersdorf discussed a class of quasi-linear Riemann-Hilbert problems for holomorphic functions and proved the existence of solutions to the problems in the Hardy space. In paper [4], moreover, the authors studied a class of quasi-linear Riemann-Hilbert problems for general holomorphic functions in similar manner and obtained the solvability of the problems in the hardy class H2 space. On the basis of the previous research work, in this paper a class of quasi-linear RiemannHubert problems for system of first order elliptic equations with general form were further discussed. Under suitable hypotheses and by means of integral operator theories, function theoretic approaches and fixed point theorem, and proved that the boundary value problems were also solvable in the corresponding functional space.