发表机构
University of Holguín; Autonomous University of Guerrero（奥尔金大学; 格雷罗自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单位球中非交换椭圆算子的Dirichlet问题，给出可解性的充要条件，并证明解的正则性比边界数据低一阶。
AI 中文摘要
本文建立了单位球$\mathbb{B}$中二阶系统$\partial_{\underline{x}} f\partial_{\underline{x}}=0$的Dirichlet问题可解性的充分必要条件，其中$\partial_{\underline{x}}$表示$\mathbb{R}^m$中Clifford代数取值的Dirac算子。我们证明，当边界数据属于$C^1(\partial\mathbb{B})$时，该问题总有解。反之，我们构造了显式反例，表明若边界不满足此光滑性假设，可解性可能失效。此外，与标准交换情形不同，我们展示了当边界数据属于$C^{k,\nu}(\partial\mathbb{B})$时，解的正则性比边界数据低一阶。
英文摘要
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 $C^{k,ν}(\partial\mathbb{B})$.
Comments15 pages
