Parabolic comparison principle and quasiminimizers in metric measure spaces

TL;DR

This paper characterizes parabolic minimizers in metric measure spaces, establishes a comparison principle and uniqueness results, and highlights differences between minimizers and quasiminimizers.

Contribution

It provides new characterizations of parabolic minimizers and a comparison principle in metric measure spaces, with insights into quasiminimizers.

Findings

Comparison principle for super- and subminimizers established

Uniqueness of minimizers in boundary value problems proven

Quasiminimizers do not generally satisfy these properties

Abstract

We give several characterizations of parabolic (quasisuper)- minimizers in a metric measure space equipped with a doubling measure and supporting a Poincar\'e inequality. We also prove a version of comparison principle for super- and subminimizers on parabolic space-time cylinders and a uniqueness result for minimizers of a boundary value problem. We also give an example showing that the corresponding results do not hold, in general, for quasiminimizers even in the Euclidean case.