Dirichlet边界条件下带边黎曼流形上p-调和函数的Yau型梯度估计
Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition
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中文总结 AI 辅助
本文在带边黎曼流形上，利用边界径向障碍与内部最大值原理，将Yau型梯度估计推广至p-调和函数，并导出Liouville定理。
中文摘要 AI 辅助
本文证明了在具有紧边界的完备黎曼流形上，在黎曼流形满足Ricci下界且边界满足平均曲率下界的条件下，对于正p-调和函数，在Dirichlet条件和外法向导数符号条件下，成立Yau型梯度估计。该结果将Kunikawa和Sakurai关于调和函数的估计推广到p-Laplace算子的完整参数范围，并在曲率假设非负时导出Liouville定理。当指数不等于2时，线性理论的截断方法不能直接推广，因为它需要到边界距离的方向Hessian，而Laplacian比较定理无法控制该量。本文在边界处用径向障碍函数替代该截断，并在内部使用内在最大值原理。
英文摘要
A Yau-type gradient estimate is proved for positive $p$-harmonic functions on complete Riemannian manifolds with compact boundary, under a Ricci lower bound on the Riemannian manifold and a mean curvature lower bound on the boundary, assuming the Dirichlet condition and a sign condition on the outward normal derivative. The result extends the estimate for harmonic functions by Kunikawa and Sakurai to the full range of $p$-Laplace operators and yields a Liouville theorem when the curvature hypotheses are nonnegative. The cutoff method of the linear theory does not extend when the exponent differs from two, since it requires a directional Hessian of the distance to the boundary that Laplacian comparison cannot control. That cutoff is replaced here by a radial barrier at the boundary and an intrinsic maximum principle in the interior.
发表机构
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
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