发表机构
Chouaib Doukkali University（舒艾布·杜卡利大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在紧致黎曼流形上研究求积曲面自由边界问题，利用变分框架和RC-GNP条件，证明了问题存在严格包含全凸包解当且仅当积分条件成立，推广了欧几里得结果并给出球面实例。
AI 中文摘要
我们将Barkatou关于求积曲面的近期结果推广到紧致黎曼流形的设定中。遵循Djité和Seck近期发展的几何与变分框架，我们将求积曲面自由边界问题表述为紧致黎曼流形上的形状优化问题。利用\\(\cite{DjiteSeck2026}\\)中引入的黎曼\\(RC\\)-GNP条件及其中建立起来的稳定性结果，我们证明：求积曲面问题\\(QS(f,k)\\)存在一个严格包含\\(f\\)的支撑集的全凸包\\(C\\)的解，当且仅当以下积分条件成立：\\(\int_C f(x)\\,dv(g) > k |\partial C|_g\\)，其中\\(dv(g)\\)是黎曼体积元，\\(|\partial C|_g\\)是\\(C\\)关于度量\\(g\\)的周长。这项工作通过将欧几里得空间替换为黎曼流形，扩展了Barkatou等人（2005）的结果。我们还提供了在标准球面上可以显式验证该条件的具体例子。本文通过建立符合Barkatou欧几里得结果精神的精确充要条件，补充了近期工作\\(\cite{DjiteSeck2026}\\)和\\(\cite{DjiteSeck2026b}\\)。
英文摘要
We study the quadrature surface free boundary problem on a compact Riemannian manifold $(\M,g)$ of dimension $N \geq 2$. Given a positive source $f \in L^2_g(\M)$ with compact support $K$, and $k>0$, one seeks a domain $Ω\supset K$ on which the overdetermined problem $-Δ_g u = f$ in $Ω$, $u=0$ and $|\nabla_g u|_g = k$ on $\partialΩ$ admits a solution. Following the geometric and variational framework of Djité--Seck, we prove that the integral condition \[ (NS)_g \qquad \int_C f\,dv(g) > k\,|\partial C|_g, \] where $C$ is the totally convex hull of $K$, is \emph{sufficient} for the existence of a solution strictly containing $C$. We then show, by an explicit counterexample on the round sphere $S^n_R$, that $(NS)_g$ is \emph{not necessary} in general under nonnegative curvature: the phenomenon arises because the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis---perimeter monotonicity within a normal convex neighborhood of $C$---under which necessity is restored, and we discuss the critical equality case $\int_C f = k|\partial C|_g$.
