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

N. V. Krylov

paper

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

arXiv:2204.13255

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 $c \geq 0$ 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 $λ u - Lu = f$ in the whole space for any $λ > 0$ . Weak uniqueness of the martingale problem associated with such operators is also discussed.