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 ut = Δu + (p−2)|∇ u|−2∇2u(∇ u,∇ u), 1<p<∞, 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 ℝn/(Lℤ)n as L → ∞. 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 Ric ≥ −κ.
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BibTeX
@misc{banerjee2026nonlinear,
title = {{A nonlinear Li-Yau inequality and its consequences}},
author = {Agnid Banerjee and Nicola Garofalo and Hamidreza Mahmoudian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.28881},
eprint = {2608.28881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28881},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A nonlinear Li-Yau inequality and its consequences
AU - Agnid Banerjee
AU - Nicola Garofalo
AU - Hamidreza Mahmoudian
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2608.28881
UR - https://arxiv.org/abs/2608.28881
AB - We prove a sharp nonlinear analogue of the Li-Yau inequality for positive solutions of the normalized parabolic p-Laplacian equation u_t = Δu + (p−2)|∇ u|^−2∇^2u(∇ u,∇ u), 1<p<∞, 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 ℝ^n/(Lℤ)^n as L → ∞. 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 Ric ≥ −κ.
ER -