Harnack's Inequality for Parabolic De Giorgi Classes in Metric Spaces

TL;DR

This paper establishes a scale-invariant Harnack inequality for parabolic De Giorgi classes in metric measure spaces, extending classical PDE results to more general non-smooth settings.

Contribution

It introduces a new framework for parabolic De Giorgi classes in metric spaces and proves a Harnack inequality applicable to these classes and quasiminimizers.

Findings

Proves local boundedness of functions in parabolic De Giorgi classes

Establishes a scale and location invariant Harnack inequality

Results apply to parabolic quasiminimizers in metric spaces

Abstract

In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincare' inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale and location invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers.