Zvonkin's transform and the regularity of solutions to double divergence form elliptic equations

TL;DR

This paper investigates the regularity and qualitative properties of solutions to double divergence form elliptic equations, establishing Harnack inequalities, new estimates, and generalizations of classical theorems using a novel Zvonkin's transform approach.

Contribution

It introduces a new analytic Zvonkin's transform for the drift coefficient, extending regularity results and existence theorems for elliptic equations with less regular coefficients.

Findings

Harnack inequality holds under Dini mean oscillation and integrability conditions.

New $L^p$-norm estimates for solutions are established.

Generalization of Hasminskii's theorem for probability solutions with Dini or VMO coefficients.

Abstract

We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on~$\mathbb{R}^d$. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix $A$ is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient $b$ is locally integrable to a power $p>d$. We establish new estimates for the $L^p$-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix $A$ satisfies Dini's condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin's transform of the drift coefficient.