Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non-smooth coefficients

We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. 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.

Automated summary

We establish surface and volume mean value formulas for uniformly parabolic equations with coefficients of low regularity. Using these formulas, we prove the parabolic strong maximum principle and the parabolic Harnack inequality. Notably, our results are based solely on classical theory and our arguments are similar to those used in the original theory of harmonic functions. We provide two proofs, relying on two different formulations of the divergence theorem, one for sets with almost C-1 boundary and the other for sets with finite perimeter.