The Harnack inequality and related properties for solutions to elliptic and parabolic equations with divergence-free lower-order coefficients

TL;DR

This paper investigates how the regularity and integrability conditions of divergence-free lower-order coefficients influence key qualitative properties of solutions to elliptic and parabolic equations, such as maximum principles and Harnack's inequality.

Contribution

It provides new criteria in Lebesgue and Morrey spaces that guarantee qualitative properties of solutions despite irregular lower-order coefficients.

Findings

Established conditions under Lebesgue spaces ensuring maximum principles.

Derived criteria in Morrey spaces for Harnack's inequality.

Extended classical results to equations with less regular coefficients.

Abstract

We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, Harnack's inequality, Liouville's theorem. The answers are given in terms of the Lebesgue spaces and the Morrey spaces.