求积曲面自由边界问题的黎曼扩展：稳定性与最优性
A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality
- Ecole Doctorale de Mathématiques et Informatique U.C.A.D.（达喀尔大学数学与信息博士学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究将求积曲面自由边界问题从欧氏环境扩展至紧黎曼流形，建立其紧性、狄利克雷问题稳定性、一阶最优性条件及相关比较原理，为该框架的黎曼扩展提供了严格依据。
AI中文摘要：
我们研究光滑紧有限维黎曼流形$(M,g)$上的求积曲面自由边界问题，该问题被构建为涉及拉普拉斯-贝尔特拉米算子狄利克雷问题的形状优化问题，带有自由边界的几何条件。在容许类满足适当均匀几何假设的条件下，我们建立该问题的紧性，并证明对应狄利克雷问题的稳定性，包括相关状态在$H_0^1(M)$中的强收敛性。我们推导对应形状泛函的一阶最优性条件，最终建立接触点处切边界的第二基本形式与平均曲率的黎曼比较原理。这些结果为求积曲面自由边界框架从欧氏环境向紧黎曼流形的扩展提供了严格依据。
英文摘要：
We study a quadrature surface free boundary problem on a smooth compact finite-dimensional Riemannian manifold $(M,g)$. The problem is formulated as a shape optimization problem involving a Dirichlet problem for the Laplace--Beltrami operator, with a geometric condition on the free boundary. Under suitable uniform geometric assumptions on the admissible class, we establish its compactness and prove the stability of the corresponding Dirichlet problems, including strong convergence of the associated states in $H_0^1(M)$. We derive the first-order optimality condition for the associated shape functional. Finally, we establish a Riemannian comparison principle for the second fundamental forms and the mean curvatures of tangent boundaries at a contact point. These results provide a rigorous extension of the quadrature surface free boundary framework from the Euclidean setting to compact Riemannian manifolds.
