On the Harnack inequality for parabolic minimizers in metric measure spaces

TL;DR

This paper establishes a scale-invariant Harnack inequality for parabolic minimizers in metric measure spaces with doubling measures and Poincaré inequalities, extending classical results to a broader geometric setting.

Contribution

It introduces a variational approach to prove Harnack inequalities for parabolic equations in metric spaces, generalizing previous Euclidean-based results.

Findings

Proves a scale-invariant Harnack inequality for minimizers of variational problems.

Extends the Grigor'yan--Saloff-Coste theorem to general p > 1 in metric spaces.

Demonstrates the effectiveness of a variational method in non-smooth geometric contexts.

Abstract

In this note we consider problems related to parabolic partial differential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincar\'e inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the sufficiency of the Grigor'yan--Saloff-Coste theorem for general p > 1 in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.