发表机构
Chouaib Doukkali University（舒艾卜杜卡利大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在黎曼几何中推广了 $p$-Laplacian 正交曲面自由边界问题存在性的充要条件，通过形状优化方法证明了在积分条件满足时解存在，并给出球面上的显式例子。
AI 中文摘要
我们将正交曲面自由边界问题存在性的充要条件推广到黎曼几何中 $p$-Laplacian 算子的情形。遵循 Barkatou 针对欧几里得情形的形状优化方法以及 Djité 和 Seck 最近发展的黎曼框架，我们将 $p$-Laplacian 正交曲面问题表述为紧致黎曼流形上的形状优化问题。利用黎曼 $RC$-GNP 条件、稳定性结果以及来自 \n\cite{DjiteSeck2026b} 的 $p$-Laplacian 形状导数公式，我们证明了 $p$-Laplacian 正交曲面问题 $QS_p(f,k)$ 在适当正则性和稳定性假设下，对于 $1 < p < \infty$，存在一个严格包含 $f$ 的支撑集的全凸包 $C$ 的解，当且仅当 \\[ \int_C f(x)\\,dv(g) > k^{p-1} |\partial C|_g. \\] 这推广了 Barkatou 在 $p=2$ 时的欧几里得结果以及 Djité 和 Seck 的黎曼结果到非线性 $p$-Laplacian 情形。我们提供了球面上的显式例子，包括径向 $p$-调和解和边界通量的计算。
英文摘要
We study the $p$-Laplacian quadrature surface free boundary problem on a compact Riemannian manifold $(\M,g)$ of dimension $N \geq 2$, for $1<p<\infty$. Given a nonnegative source $f \in L^p_g(\M)$ with compact support $K$, and $k>0$, one seeks a domain $Ω\supset K$ on which the overdetermined problem $-Δ_{p,g} u = f$ 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 \[ (NS)_{p,g} \qquad \int_C f\,dv(g) > k^{p-1}\,|\partial C|_g, \] 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 $(NS)_{p,g}$ is \emph{not necessary} in general under nonnegative curvature: the mechanism is that the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis---perimeter monotonicity on the admissible class inside a normal convex neighborhood of $C$---under which necessity is restored, and we discuss the critical equality case $\int_C f\,dv(g) = k^{p-1}|\partial C|_g$.
