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

TL;DR

This paper proves H"older continuity for solutions to a class of anisotropic degenerate parabolic equations using intrinsic Harnack estimates and explores their equivalent formulations within the intrinsic geometry.

Contribution

It introduces a new proof of H"older continuity for solutions to anisotropic equations and establishes equivalent forms of intrinsic Harnack estimates.

Findings

Proved H"older continuity for solutions under specified conditions.

Established equivalent formulations of Harnack estimates in intrinsic geometry.

Extended understanding of anisotropic parabolic equations.

Abstract

We give a proof of H\"older 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.