Year: 2026
Type: preprint
Abstract: In this paper we establish a necessary and sufficient condition for the solvability of the Dirichlet problem in the unit ball $\mathbb{B}$ for the second order system $\partial_{\underline{x}} f\partial_{\underline{x}}=0$, where $\partial_{\underline{x}}$ stands for the Clifford-algebra valued Dirac operator in $\mathbb{R}^m$. We prove that the problem admits a solution whenever the boundary data belong to $C^1(\partial\mathbb{B})$. Conversely, we construct explicit counterexamples showing that solvability may fail if this smoothness assumption on the boundary is not satisfied. Moreover, in contrast with the standard commutative setting, we show how that solutions exhibit a regularity one order less than the boundary data, when the latter belong to $C^{k,ν}(\partial\mathbb{B})$.
Source: arXiv (Cornell University)
Institutions: University of Holguín, Universidad Autónoma de Guerrero
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212083442