Liouville rigidity and time-extrinsic Harnack estimates for an anisotropic slow diffusion

Source abstract

We prove that non-negative solutions to the fully anisotropic equation ∂ t u = ∑ i = 1 N ∂ i ( | ∂ i u | p i − 2 ∂ i u ) , in R N × ( − ∞ , T ) , $$\begin{equation*} \partial _t u= \sum _{i=1}^N \partial _i (|\partial _i u|^{p_i-2} \partial _i u), \quad \qquad \text {in } \mathbb {R}^N\times (-\infty , T), \end{equation*}$$ are constant if they satisfy a condition of finite speed of propagation and if they are both one-sided bounded, and bounded in R N \mathbb {R}^N at a single time level. A similar statement is valid when the bound is given at a single space point. As a general paradigm, local Hölder estimates provide the basics for rigidity. Finally, we show that recent intrinsic Harnack estimates can be improved to a Harnack inequality valid for non-intrinsic times. Locally, they are equivalent.