Linear and fully nonlinear elliptic equations with Morrey drift

TL;DR

This paper investigates the solvability and regularity of linear and fully nonlinear elliptic equations with coefficients in Morrey and VMO classes, extending known results even for equations involving the Laplacian.

Contribution

It provides new results on the regularity of solutions to elliptic equations with Morrey class coefficients, including fully nonlinear cases, broadening the understanding of such equations.

Findings

Second derivatives of solutions are in a local Morrey class containing W^{2}_{p,loc}

Results apply to equations with coefficients in Morrey and VMO classes

Extends regularity results to fully nonlinear elliptic equations

Abstract

We present some results concerning the solvability of linear elliptic equations in bounded domains with the main coefficients almost in VMO, the drift and the free terms in Morrey classes containing $L_{d}$, and bounded zeroth order coefficient. We prove that the second-order derivatives of solutions are in a local Morrey class containing $W^{2}_{p,loc}$. Actually, the exposition is given for fully nonlinear equations and encompasses the above mentioned results, which are new even if the main part of the equation is just the Laplacian.