On Holder Continuity and Equivalent Formulation of Intrinsic Harnack Estimates for an Anisotropic Parabolic Degenerate Prototype Equation

Abstract

We give a proof of Hölder continuity for bounded local weak solutions to the equation $u_t={\sum }_{i=1}^N(|u_{x_i}{|}^{p_i-2}{u}_{x_i}{)}_{x_i}$, in $\Omega \times [0,T]$, with $\Omega \subset \subset {\mathbb{R}}^N$, under the condition $2<p_i<\bar{p}(1+2/N)$ for each $i=1,..,N$, being $\bar{p}$ the harmonic mean of the $p_i$s, via recently discovered intrinsic Harnack estimates. Moreover we establish equivalent forms of these Harnack estimates within the proper intrinsic geometry.

arXiv commentHölder continuity of solutions to the anisotropic p-laplacian equation is valid in the range 2<p_i<p(1+1/N), as an indirect consequence of finite speed of propagation. For this we redirect the reader to arXiv:2211.03357, or https://hdl.handle.net/2158/1263325. Finally, we are not aware of a proof of Hölder continuity for the whole range of boundedness 2<p_i<p(1+2/N)