LIOUVILLE RIGIDITY AND TIME-EXTRINSIC HARNACK ESTIMATES FOR AN ANISOTROPIC SLOW DIFFUSION

We prove that non-negative solutions to the fully anisotropic equation N � partial differential tu = partial differential i(| partial differential iu|pi-2 partial differential iu), in RN x (-& INFIN;, T), i=1 are constant if they satisfy a condition of finite speed of propagation and if they are both one-sided bounded, and bounded in RN 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 Ho & BULL;lder 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.