Title: On Dirichlet and Neumann type problems of polynomial Dirac equations with boundary conditions Authors: Constales, DenisGrob, DennisKrausshar, Rolf Sören ×Sprössig, Wolfgang # Issue Date: 2009 Publisher: John Wiley and Sons Ltd. Series Title: Mathematical Methods in the Applied Sciences issue:submitted Abstract: Let ${\bf D}_{\bf x}:= \sum_{i=1}^n \frac{\partial }{\partial x_i} e_i$ be the Euclidean Dirac operator in $\R^n$ and let $P(X) = a_m X^m + \ldots + a_1 X_1 + a_0$ be a polynomial with arbitrary complex coefficients. Differential equations of the form $P({\bf D}_{\bf x})u({\bf x}) = 0$ are called homogeneous polynomial Dirac equations with complex coefficients. In this paper we treat Dirichlet and Neumann type problems of the general form $P({\bf D}_{\bf x}) u({\bf x}) = f({\bf x})$ with prescribed boundary conditions that avoid blow-ups inside of 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, Helmholtz and the Klein-Gordon equation are included as special subcases in this context. ISSN: 0170-4214 Publication status: submitted KU Leuven publication type: IT Appears in Collections: Analysis Section