有界区域上超解析函数的Carleman型边值问题

Carleman Type Boundary Problem of the Hyperanalytic Functions in the Bounded Domains

在平面和薄壳弹性理论中遇到的某些偏微分方程组可以化为Douglis所研究过的那一类一阶椭圆组,在Douglis代数(e^r=0,λ∈N;ea=ae,a∈C)下可写成简单的形式 DW=0, (1)其中是Douglis微分算子,是复平面到这个代数的映射,为复值函数。 方程(1)的正则解称为超解析函数。对超解析函数的各种边值问题的研究已有许多结果。本文研究超解析函数的Carleman型边值问题,建立了解的表示式,并得到了线性无关解的个数及可解条件的个数与问题的指数之间的关系。为简单计,本文直接引用文中的记号和术语,不再复述。

In this paper we consider the Carleman type boundary value problem for a class of the hypercomplex equations. Solutions such equations are termed heperanalytic function. An explicit form of solution is found in the case of finitsimply-connected domains. We also obtain necessary and sufficient conditions for the solvability of the boundary value problem and relationship between solutions and jndies of the problem.