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

Emanuele Malagoli, Diego Pallara and Sergio Polidoro

Mathematische Nachrichten, 2023, vol. 296, issue 9, 4236-4263

Abstract: 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 C1‐boundary, the other stated for sets with finite perimeter.

Date: 2023
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https://doi.org/10.1002/mana.202100612

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