Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients

TL;DR

This paper establishes a Harnack inequality for nonnegative solutions of certain parabolic equations in double divergence form, under conditions on coefficients that include Dini mean oscillation and Morrey class assumptions.

Contribution

It provides new theoretical results for parabolic equations with singular coefficients in double divergence form, extending existing inequalities to broader coefficient classes.

Findings

Harnack inequality proven for equations with Dini mean oscillation coefficients

Results applicable to Fokker-Planck-Kolmogorov equations

Advances the theoretical understanding of parabolic equations with singular coefficients

Abstract

This paper investigates the Harnack inequality for nonnegative solutions to second-order parabolic equations in double divergence form. We impose conditions where the principal coefficients satisfy the Dini mean oscillation condition in $x$, while the drift and zeroth-order coefficients belong to specific Morrey classes. Our analysis contributes to advancing the theoretical foundations of parabolic equations in double divergence form, including Fokker-Planck-Kolmogorov equations for probability densities.