非线性抛物方程的Li-Yau和Harnack估计
Li-Yau and Harnack estimates for nonlinear parabolic equations
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中文总结 AI 辅助
本文为多类非线性抛物方程建立全局Harnack不等式，包括一致抛物流、加权p-Laplacian热流及各向同性扩散，利用极大值原理和加权测地线变分，并在粘性意义下推导Li-Yau不等式。
中文摘要 AI 辅助
我们为几类非线性抛物方程的正粘性解建立了全局Harnack不等式。这些方程包括欧几里得空间和平坦环面上的正齐次一致抛物流、归一化加权$p$-Laplacian热流，以及闭光滑度量测度空间上的齐次各向同性扩散。对于归一化无穷Laplacian热流，不需要曲率假设。对于普通加权$p$-Laplacian，我们在非负Bakry-Émery Ricci下界条件下获得了压力估计。证明结合了在半连续函数的极大值原理下两个独立时空点的比较与加权测地线变分。相应的Li-Yau不等式随后在一阶粘性意义下成立。
英文摘要
We establish global Harnack inequalities for positive viscosity solutions of several classes of nonlinear parabolic equations. The equations include positively homogeneous uniformly parabolic flows on Euclidean space and flat tori, normalized weighted $p$-Laplacian heat flows, and homogeneous isotropic diffusions on closed smooth metric measure spaces. For the normalized infinity-Laplacian heat flow, no curvature assumption is needed. For the ordinary weighted $p$-Laplacian, we obtain pressure estimates under nonnegative Bakry-Émery Ricci lower bounds. The proofs combine comparisons at two independent space-time points using the maximum principle for semicontinuous functions with weighted geodesic variations. The corresponding Li-Yau inequalities then follow in the first-order viscosity sense.
发表机构
- Wichita State University（威奇托州立大学）
- Auburn University（奥本大学）
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