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arXiv · 2312.06404

Harnack inequality and the relevant theorems on Finsler metric measure manifolds

Xinyue Cheng·Yalu Feng
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Abstract

In this paper, we carry out in-depth research centering around the Harnack inequality for positive solutions to nonlinear heat equation on Finsler metric measure manifolds with weighted Ricci curvature ${\rm Ric}_{\infty}$ bounded below. Aim on this topic, we first give a volume comparison theorem of Bishop-Gromov type. Then we prove a weighted Poincar\'{e} inequality by using Whitney-type coverings technique and give a local uniform Sobolev inequality. Further, we obtain two mean value inequalities for positive subsolutions and supersolutions of a class of parabolic differential equations. From the mean value inequality, we also derive a new local gradient estimate for positive solutions to heat equation. Finally, as the application of the mean value inequalities and weighted Poincar\'{e} inequality, we get the desired Harnack inequality for positive solutions to heat equation.

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Xinyue Cheng, Yalu Feng. 2023-12-11. Harnack inequality and the relevant theorems on Finsler metric measure manifolds. https://arxiv.org/abs/2312.06404

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