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(2013)RESULTS IN MATHEMATICS.64(1-2).p.193-213 Mark 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 $P(X) = a_m X^m + \ldots + a_1 X + a_0$ be a polynomial with real coefficients. Differential equations of the form $P({\bf D}_{\bf x})u({\bf x}) = 0$ are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slighly less general form $P({\bf D}_{\bf x}) u({\bf x}) = f({\bf x})$ (where the roots are exclusively real) 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 and the Klein Gordon equation are included as special subcases in this context.

Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-4160361



Author: Denis Constales , Dennis Grob and Rolf Soeren Krausshar

Source: https://biblio.ugent.be/publication/4160361



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