发表机构
Escola Politécnica de Pernambuco, Universidade de Pernambuco; Departamento de Matemática, Universidade Federal de Campina Grande（伯南布哥联邦大学理学院; 坎皮纳格兰德联邦大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对 $p$-Laplacian 正解建立参数化 Heintze--Karcher 型不等式，并利用亏量恒等式改进条件，进而获得度量球与径向解的刚性分类。
AI 中文摘要
我们建立了有界黎曼区域上涉及 $p$-Laplacian（$p\geq2$）的非线性 Dirichlet 问题的正解的参数化 Heintze--Karcher 型不等式。在 Ricci 曲率下界、正边界平均曲率以及合适的结构和逼近假设下，该估计由正则化的 Reilly 型恒等式和加权 Hessian 不等式得出。我们将所得界与先前的估计进行比较，并研究等式成立时的相关几何。对于 $p=2$，一个精确的亏量恒等式允许将逐点条件 $f'\leq nk$ 替换为加权积分条件。作为应用，我们获得了 Heintze--Karcher 和肥皂泡型刚性结果，其中等式迫使区域为度量球且解为径向的。
英文摘要
We establish a parameterized Heintze--Karcher-type inequality for positive solutions of nonlinear Dirichlet problems involving the $p$-Laplacian, with $p\geq2$, on bounded Riemannian domains. Under a Ricci curvature lower bound, positive boundary mean curvature, and suitable structural and approximation assumptions, the estimate follows from a regularized Reilly-type identity and a weighted Hessian inequality. We compare the resulting bound with previous estimates and investigate the geometry associated with equality. For $p=2$, an exact deficit identity allows the pointwise condition $f'\leq nk$ to be replaced by a weighted integral condition. As applications, we obtain Heintze--Karcher and Soap Bubble-type rigidity results, with equality forcing the domain to be a metric ball and the solution to be radial.
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