A nonlinear Li-Yau inequality and its consequences

Abstract

We prove a sharp nonlinear analogue of the Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation \[ u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u),\ \ \ 1<p<\infty, \] on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpreted in the viscosity sense on the critical set of the solution. We also derive a sharp global Harnack inequality from the gradient estimate. The constant is best possible. Equality holds identically for an explicit self-similar solution in $\Rn$, and sharpness on closed manifolds follows by taking the limit along the flat tori $\mathbb R^n/(L\mathbb Z)^n$ as $L \to \infty$. The proof rests on an exact Bochner-type identity for a family of uniformly parabolic approximating flows. We choose the first-order term separately from the second-order regularization so that the maximum principle yields the sharp constant, uniformly in the regularization parameter and without any assumption on the critical set of the solution. Using this Bochner identity with a modified Li-Yau functional, we extend the estimate to closed manifolds with $\operatorname{Ric} \ge -\kappa$.