Quantitative regularity for parabolic De Giorgi classes

TL;DR

This paper develops a quantitative version of the De Giorgi method for establishing interior Hölder regularity of solutions to parabolic equations with rough coefficients, extending previous non-quantitative approaches.

Contribution

It introduces a quantitative approach to the parabolic intermediate value lemma within the De Giorgi method for parabolic classes, enhancing regularity analysis.

Findings

Established a quantitative interior Hölder regularity estimate.

Extended De Giorgi method to parabolic classes with rough coefficients.

Provided a quantitative version of the intermediate value lemma.

Abstract

We deal with the De Giorgi H{\"o}lder regularity theory for parabolic equations with rough coefficients and parabolic De Giorgi classes which extend the notion of solution. We give a quantitative proof of the interior H{\"o}lder regularity estimate for both using De Giorgi method. Recently, the De Giorgi method initially introduced for elliptic equation has been extended to parabolic equation in a non quantitative way. Here we extend the method to the parabolic De Giorgi classes in a quantitative way. To this aim, we get a quantitative version of the non quantitative step of the method, the parabolic intermediate value lemma, one of the two main tools of the De Giorgi method sometimes called ``second lemma of De Giorgi''.