Riemann boundary value problem for hyperanalytic functions

The theory of Riemann boundary value problem for analytic functions of one complex variable and singular integral equations that are equivalent to it has been extensively studied in the literature. For classical books on this topic see [7, 12, 13] and for an actual overview of them the reader is directed to the monograph by Estrada and Kanwal [6], and the references therein. In the more recent times several generalizations and extensions of the theory are treated and have led to numerous important results not only for nonsmoothly bounded domain, which differs with the former, but for general assumptions on the data of the problem, such as generalized Hölder coefficients or special subspaces of this space and the desired boundary behavior condition for the solution. During the last decades, the Riemann boundary value problem was studied for generalized analytic functions, as well as for many other linear and nonlinear elliptic systems in the plane [1, 2, 8, 15, 16, 17]. Let γ be a rectifiable positively oriented Jordan closed curve with diameter d which is the boundary of a bounded simply connected domainΩ+ in the complex plane C and let Ω− := C \ (Ω+∪ γ). In the Douglis commuting function algebra sense, a continuously differentiable null solution to the Douglis differential operator provides us with the class of hyperanalytic functions. Let (Ω±) be the spaces of all continuous functions on Ω± := Ω± ∪ γ and hyperanalytic in Ω±. The classical Riemann boundary value problem for analytic functions consists in finding a function Φ(z) analytic in C \ γ, such that Φ has a finite order at infinity, and satisfies a prescribed jump condition across the curve γ. The basic boundary condition takes