Mean value formulas for classical solutions to uniformly parabolic equations in divergence form

TL;DR

This paper establishes mean value formulas for classical solutions of uniformly parabolic divergence form equations, leading to proofs of the strong maximum principle and Harnack inequality using classical harmonic function methods.

Contribution

It introduces surface and volume mean value formulas for parabolic equations and derives key principles without relying on advanced modern techniques.

Findings

Mean value formulas for classical solutions

Proofs of the strong maximum principle and Harnack inequality

Methods based on classical divergence theorem formulations

Abstract

We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C^1 boundary, the other stated for sets with finite perimeter.