非线性Li-Yau不等式及其推论
A nonlinear Li-Yau inequality and its consequences
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中文总结 AI 辅助
该研究针对非负Ricci曲率闭黎曼流形上的归一化抛物型p-拉普拉斯方程，证明了尖锐非线性Li-Yau不等式，推导了相关推论及尖锐全局Harnack不等式。
中文摘要 AI 辅助
我们证明了著名的Li-Yau不等式的一个尖锐非线性版本，适用于非负Ricci曲率的闭黎曼流形上的归一化抛物型p-拉普拉斯方程 $u_t = \nabla u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$（其中 $1<p<\infty$）的正解，该方程在解的临界集上按粘性意义解释。该常数是最优的，即使在闭流形类中也是如此：它在平坦 $\mathbb{R}^n$ 中被一个显式自相似解恒等地达到，并且通过平坦环面 $\mathbb{R}^n/(L\mathbb{Z})^n$ 沿 $L \to \infty$ 的大环面极限，其尖锐性可传递到紧流形情形。证明依赖于一类一致抛物近似流的精确Bochner型恒等式，在这类流中，二阶正则项与一阶 eikonal 项是解耦的：对于该类流，最大值原理以最优常数成立，且在正则参数范围内一致，无需对解的临界集做任何假设。此外，该恒等式在Li-Yau泛函的经典α-松弛下是形式不变的；因此，我们也得到了Ricci曲率满足 $\operatorname{Ric} \ge -\kappa$ 的闭流形上的对应不等式（同样仅需Ricci下界的假设），以及在Ricci曲率非负的完备非紧流形上，近似流的尖锐不等式——同样无需对临界集做任何假设——只要满足距离函数的截断假设（在 $\mathbb{R}^n$ 中自动成立，且当 $p \ge 2$ 时在非负截面曲率下也成立）和Li-Yau量的定性多项式增长条件（极值剖面满足指数为零的该条件）。由此可推出一个尖锐的全局Harnack不等式。
英文摘要
We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = Δu + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpreted in the viscosity sense on the critical set of the solution. The constant is best possible, even within the class of closed manifolds: it is attained identically by an explicit self-similar solution in flat $\Rn$, and its sharpness transfers to the compact setting through a large-torus limit along the flat tori $\mathbb R^n/(L\mathbb Z)^n$, $L \to \infty$. The proof rests on an exact Bochner-type identity for a family of uniformly parabolic approximating flows in which the second-order regularization and the first-order eikonal term are decoupled: for this family the maximum principle applies with the sharp constant, uniformly in the regularization parameter, and with no assumption on the critical set of the solution. The identity is moreover form-invariant under the classical $α$-relaxation of the Li-Yau functional; as a consequence we also obtain the corresponding inequality on closed manifolds with $\operatorname{Ric} \ge -κ$ (again with no assumption beyond the Ricci lower bound), and, on complete noncompact manifolds with $\operatorname{Ric} \ge 0$, the sharp inequality for the approximating flows -- again with no assumption on the critical set -- under a cutoff hypothesis on the distance function (automatic in $\Rn$, and, for $p \ge 2$, under nonnegative sectional curvature) and a qualitative polynomial growth condition on the Li-Yau quantity, satisfied, with exponent zero, by the extremal profile. A sharp global Harnack inequality follows.
