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The Dirichlet problem for a non-commutative elliptic operator in the ball
Abstract
In this paper we establish a necessary and sufficient condition for the solvability of the Dirichlet problem in the unit ball 𝔹 for the second order system ∂underlinex f∂underlinex=0, where ∂underlinex stands for the Clifford-algebra valued Dirac operator in ℝm. We prove that the problem admits a solution whenever the boundary data belong to C1(∂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 Ck,ν(∂𝔹).
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BibTeX
@misc{garcia2026dirichlet,
title = {{The Dirichlet problem for a non-commutative elliptic operator in the ball}},
author = {A. Moreno García and D. Alfonso Santiesteban and R. Abreu Blaya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07770},
eprint = {2609.07770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07770},
}
RIS
TY - UNPB
TI - The Dirichlet problem for a non-commutative elliptic operator in the ball
AU - A. Moreno García
AU - D. Alfonso Santiesteban
AU - R. Abreu Blaya
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07770
UR - https://arxiv.org/abs/2609.07770
AB - In this paper we establish a necessary and sufficient condition for the solvability of the Dirichlet problem in the unit ball 𝔹 for the second order system ∂_underlinex f∂_underlinex=0, where ∂_underlinex stands for the Clifford-algebra valued Dirac operator in ℝ^m. We prove that the problem admits a solution whenever the boundary data belong to C^1(∂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(∂𝔹).
ER -