Riemann-Hilbert problems for monogenic functions in axially symmetric domains

The classic theory of Riemann-Hilbert boundary value problems (for short RHBVPs) for analytic functions is closely connected with the theory of singular integral equations and has a wide range of applications in other fields, such as the theory of elasticity, quantum mechanics, statistical physics, the theory of orthogonal polynomials, and asymptotic analysis (cf. [1–7]). Over the years this type of boundary value problems has been systematically investigated by many authors, e.g., [1–4, 8–10]. Therefore, it is natural to attempt to generalize this type of boundary value problem to higher dimensions. This will be not only a purely theoretical question, since such problems are closely linked to the physical applications like transport problems or problems in hydro-dynamic mechanics.

To our knowledge, there are two principal ways to generalize complex analysis to higher dimensions, the theory of several complex variables and Clifford analysis. The latter, in particular quaternionic analysis, is the theory of so-called monogenic functions and a refinement of real harmonic analysis in the sense that the principal operator in Clifford analysis, the Dirac or generalized Cauchy-Riemann operator, factorizes the higher Laplace operator. As principal references for this theory we mention [11–14]. Furthermore, in the context of Clifford analysis RHBVPs, particularly Riemann boundary value problems, with constant coefficients were widely discussed; see, e.g., [15–22]. Here, a direct application of the properties of the Cauchy integral operator and existing power series expansions allows one to represent the solution in terms of integral operators and Taylor series expansions. This kind of Riemann boundary value problems is also closely connected with other types of boundary value problems for different partial differential equations in higher dimensions; see, e.g., [23–26]. However, there are almost no results with respect to RHBVPs with variable coefficients even after more than 30 years of research in this direction. The obstacles lie in the following facts: while monogenic functions retain many of the properties of analytic function, there are two drawbacks. First, the product of monogenic functions is in general not monogenic and second the composition of monogenic is in general not monogenic. While the latter property is not so important for our problem at hand the former is a major problem. Moreover, even in the simplest case of quaternionic analysis, we have several major problems, such as the lack of a function which has the same properties as the logarithmic function of one complex variable. These make it very difficult to solve RHBVPs with variable coefficients for monogenic functions in higher dimensions by directly employing classic methods from complex analysis. But, as a particular interesting case, it was first introduced by Fueter in the case of quaternionic analysis in [11] that the product of monogenic functions of axial type is still a monogenic function of axial type, and later on this class was studied in detail by Sommen et al. (cf. [27, 28]). This makes this class more similar to the classic case of analytic functions in the plane and, therefore, if one wants to generalize any property or fact from complex analysis to quaternionic or Clifford analysis, the standard approach is to see if it can be done for axially monogenic functions. So, instead of trying to solve this problem directly, we are going to show that RHBVPs with variable coefficients for monogenic functions with axial symmetry can be solved in the context of quaternionic analysis. This represents an initial and very crucial step to solve RHBVPs for monogenic functions in the future.

Our idea is to transform a RHBVP with variable coefficients for monogenic functions with axial symmetry in \(\mathbb{R}^{4}\) into a RHBVP for analytic functions over the complex plane, use the classic results from complex analysis, and transfer it back. In this way we can obtain the solution in an explicit form. As a special case we present the solution to the Schwarz problem for monogenic functions with axial symmetry. Furthermore, we adapt the approach above and generalize to the case of null-solutions to \((\mathcal{D}-\alpha)\phi=0\), \(\alpha \in\mathbb{R}\), where \(\mathbb{R}\) denotes the field of real numbers (see Section 3.3). Here, the main contribution that we emphasize lies in constructing a way to solve a RHBVP with variable coefficients in four-dimensional spaces. Moreover, by using the possibility to decompose a monogenic functions into a sum of monogenic functions with axial symmetry, we plan to extend this idea to the case of RHBVPs with variable coefficients for monogenic functions.

The paper is organized as follows. In Section 2, we will recall the necessary facts about quaternion analysis. In Section 3, we will focus on RHBVPs with variable coefficients for monogenic functions with axial symmetry. We will derive the solution to the RHBVP with boundary data belonging to a Hölder space in terms of an integral representation. Moreover, we give the solutions to the Schwarz problem for monogenic functions and null-solutions to \((\mathcal {D}-\alpha)\phi=0\), \(\alpha\in\mathbb{R}\) with axial symmetry, where \(\mathbb{R}\) denotes the field of real numbers.

Good introductions to Clifford algebras, Quaternions, and Clifford analysis can be found in [12–14, 26, 27].

An arbitrary \(x\in\mathbb{H}\) can be written as \(x=x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3} \triangleq x_{0}+\underline{x}\), where \(\operatorname{Sc}(x)\triangleq x_{0}\) and \(\operatorname{Vec}(x)\triangleq \underline{x}\) are the scalar and vector part of \(x\in\mathbb{H}\), respectively. Elements \(x\in\mathbb{R}^{4}\) can be identified with quaternions \(x\in\mathbb{H}\). The conjugation is defined by \(\bar{x}=\sum^{3}_{j=0}x_{j}\bar{e}_{j}\) with \(\bar{e}_{0}=e_{0}\) and \(\bar{e}_{j}=-e_{j}\), \(j=1,2,3\), and hence, \(\overline{x y}=\bar{y}\bar{x}\). The algebra of quaternions possesses an inner product \((y,x)= \operatorname{Sc}(y\bar{x}) =(y\bar{x})_{0}\) for all \(x,y\in\mathbb{H}\).

In this paper we will consider the generalized Cauchy-Riemann operator \(\mathcal{D}=\sum^{3}_{j=0}e_{j}\partial_{x_{j}}\) in \(\mathbb{R}^{4}\). The generalized Cauchy-Riemann operator factorizes the Laplacian in the following sense: \(\overline{\mathcal{D}}\mathcal{D}=\sum^{3}_{j=0}\partial^{2}_{x_{j}} =\Delta\), where Δ denotes the Laplacian in \(\mathbb{R}^{4}\).

We also need to define axially symmetric open sets.

Let Ω be an axially symmetric domain of \(\mathbb{R}^{4}\) with smooth boundary ∂Ω. An \(\mathbb{H}\)-valued function \(\phi= \sum_{j=0}^{3} \phi _{j}e_{j} \) is continuous, Hölder continuous, p-integrable, continuously differentiable and so on if all components \(\phi_{j}\) have that property. The corresponding function spaces, considered as either right-Banach or right-Hilbert modules, are denoted by \(C(\Omega,\mathbb{H})\), \(H^{\mu}(\Omega,\mathbb{H})\) (\(0<\mu\leqslant 1\)), \(L_{p}(\Omega,\mathbb{H})\) (\(1< p<+\infty\)), \(C^{1}(\Omega,\mathbb{H})\), respectively.

Throughout this paper any functions defined on \(\Omega\cup \partial\Omega\subset\mathbb{R}^{4}\) with values in \(\mathbb{H}\) are supposed to be of axial type unless otherwise stated.

In this section we discuss RHBVPs with variable coefficients for monogenic functions. In this case we will restrict ourselves to functions which have axial symmetry and are defined over axial domains of \(\mathbb{R}^{4}\). We first solve the RHBVP in \(\mathbb{R}^{4}\) by transforming it into a RHBVP for complex analytic functions. In the case of boundary values belonging to a Hölder space we represent its solution in terms of an explicit integral representation formula. As a corollary we obtain the solution to the corresponding Schwarz problem in quaternion analysis.