Linear and fully nonlinear elliptic equations with $L_{d}$-drift

TL;DR

This paper studies elliptic equations with $L_{d}$-drift, establishing existence of solutions in certain Sobolev spaces under BMO conditions, extending known results even for linear cases.

Contribution

It proves existence of solutions in $W^{2}_{p, ext{loc}}$ for nonlinear elliptic equations with $L_{d}$-drift, a novel result even in the linear setting.

Findings

Existence of solutions in $W^{2}_{p, ext{loc}}$ for certain $p$

Results hold under BMO dependence on $x$

Applicable to both linear and nonlinear equations

Abstract

In subdomains of $\mathbb{R}^{d}$ we consider uniformly elliptic equations $H\big(v( x),D v( x),D^{2}v( x), x\big)=0$ with the growth of $H$ with respect to $|Dv|$ controlled by the product of a function from $L_{d}$ times $|Dv|$. The dependence of $H$ on $x$ is assumed to be of BMO type. Among other things we prove that there exists $d_{0}\in(d/2,d)$ such that for any $p\in(d_{0},d)$ the equation with prescribed continuous boundary data has a solution in class $W^{2}_{p,\text{loc}}$. Our results are new even if $H$ is linear.