Elliptic equations in Sobolev spaces with Morrey drift and the zeroth-order coefficients

TL;DR

This paper establishes solvability of elliptic equations with coefficients in Morrey and VMO classes in Sobolev spaces, and discusses weak uniqueness of related martingale problems, extending classical results to less regular coefficients.

Contribution

It introduces new solvability results for elliptic equations with Morrey and VMO coefficients in Sobolev spaces, and addresses weak uniqueness of the associated martingale problem.

Findings

Solvability in Sobolev spaces for elliptic equations with Morrey and VMO coefficients.

Extension of solvability results to bounded $C^{1,1}$-domains and the whole space.

Discussion of weak uniqueness for the martingale problem associated with these operators.

Abstract

We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b$ in a Morrey class containing $ L_{d}$, and $c\geq0$ in a Morrey class containing $L_{d/2}$. We prove the solvability in Sobolev spaces of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains, and of $\lambda u-Lu=f$ in the whole space for any $\lambda>0$. Weak uniqueness of the martingale problem associated with such operators is also discussed.