On Dirichlet Type Problems of Polynomial Dirac Equations with Boundary Conditions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$$\end{document} be the Euclidean Dirac operator in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document} and let P(X) = amXm + . . . + a1X +  a0 be a polynomial with real coefficients. Differential equations of the form P(Dx)u(x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P(Dx)u(x) = f(x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein–Gordon equation are included as special subcases in this context.