发表机构
Univ Rennes, INRIA, IRMAR [(Institut de Recherche Mathematique de Rennes)] - UMR 6625（雷恩大学，法国国家信息与自动化研究所，雷恩数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整刻画了一般次椭圆过程在光滑域外吸收时的正则与不规则边界点，建立经典边值问题的适定性，并揭示解在边界可能不连续，给出保持连续性的锥条件。
AI 中文摘要
我们完整描述了一类一般次椭圆随机过程 $(X(t))_{t\geq0}$ 在 $C^{3,1}$ 域 $\Omega$ 外被吸收时的正则边界点与不规则边界点集合。同时，在经典意义下建立了 $\Omega$ 上相关边值问题的适定性。需要强调的是，在整个工作中，我们未对 $\Omega$ 的边界内在结构作任何假设。特别地，特征点与非特征点可能共存于 $\partial\Omega$ 上，从而推广了文献中已有的结果。我们表明，与以往情形不同，该边值问题的解在边界 $\partial\Omega$ 处可能表现出不连续性。在某些涉及 $C^\infty$ 域 $\Omega$ 与 $C^\infty$ 系数的情况下，不连续集甚至在 $\partial\Omega$ 上具有正的 Lebesgue 曲面测度。最后，我们强调了一个锥条件，在该条件下边界处的连续性得以保持。
英文摘要
We fully describe the set of regular and irregular boundary points for a general hypoelliptic stochastic process $(X(t))_{t\geq0}$ absorbed outside a $C^{3,\,1}$ domain $Ω$. The well-posedness of the associated boundary-value problem on $Ω$ is also established in the classical sense. Let us emphasize that, throughout this work, no assumption is made regarding the inherent structure of the boundary of $Ω$. In particular, characteristic and non-characteristic points may coexist on $\partialΩ$, thus extending previous results obtained in the literature. We show that, unlike previous settings, the solution to this boundary-value problem may exhibit discontinuity at the boundary $\partialΩ$. In certain cases involving a $C^\infty$ domain $Ω$ and $C^\infty$ coefficients, the discontinuity set is even of positive Lebesgue surface measure on $\partialΩ$. Finally, we highlight a cone condition under which the continuity is preserved at the boundary.
