Riemann boundary value problems on the sphere in Clifford analysis
Clifford analysis offers an elegant generalization of analytic functions from the complex plane to higher dimensional space. It focuses on the study of so-called monogenic functions, i.e. null solutions of the Dirac operator or the generalized Cauchy-Riemann operator \(\mathcal{D}\). Starting from the polynomial \(p(\lambda) = \lambda^k + a_1 \lambda^{k-1} + \ldots + a_k\) (\(a_i \in \mathbb{C}\), \( i=1,2,\ldots,k\)), a polynomially generalized Cauchy-Riemann operator \(p(\mathcal{D})\) can be defined: \[ p(\mathcal{D}) = \mathcal{D}^k +a_1 \mathcal{D}^{k-1} + \ldots + a_k, \] where \(a_k\) means \(a_k I\), \(I\) denoting the identity operator. The solutions to polynomially generalized Cauchy-Riemann equations \(p(\mathcal{D}) \phi = 0\) are called polynomially monogenic functions. In this paper a type of Riemann boundary value problems for polynomially monogenic functions over the sphere in \(\mathbb{R}^{n+1}\) is studied. To this end the Fischer type decomposition is established for null solutions of the differential operator \((\mathcal{D} - a(x))^k\), including the polynomial generalized Cauchy-Riemann operators \((\mathcal{D} - \lambda)^k\) and \(\mathcal{D} - \lambda\) as special cases, where \(a \in C^k(\Omega, \mathbb{C}^{n+1})\) satisfies \(\mathcal{D} \gamma(x) = a(x)\) with \(\gamma \in C^{k+1}(\Omega,\mathbb{C})\), \(\Omega \subset \mathbb{R}^{n+1}\) and \(\lambda \in \mathbb{C}\). Applying this Fischer type decomposition and the Clifford calculus for polynomial monogenic functions, explicit expressions of solutions to Riemann boundary value problems for polynomially monogenic functions over the sphere of \(\mathbb{R}^{n+1}\) are obtained. As special cases, the solutions to classical Riemann boundary value problems for polyanalytic functions and metaanalytic functions in the complex plane are derived, which are important not only for pure mathematical reasons, but also for concrete problems in physics and engineering, particularly when considered over the sphere of \(\mathbb{R}^3\) and the unit circumference of \(\mathbb{R}^2\).
scientific article; zbMATH DE number 6131881
scientific article