On linear elliptic and parabolic equations with growing drift in Sobolev spaces without weights

TL;DR

This paper investigates linear elliptic and parabolic equations with unbounded drift terms, establishing existence and regularity of solutions in Sobolev spaces without weights under bounded VMO coefficients.

Contribution

It extends the theory of second-order PDEs by allowing growing drift coefficients and proving solution regularity in unweighted Sobolev spaces.

Findings

Existence of solutions with derivatives in Sobolev spaces

Regularity results for equations with growing drift

Solutions are integrable to the p-th power in Lebesgue measure

Abstract

We consider uniformly elliptic and parabolic second-order equations with bounded zeroth-order and bounded VMO leading coefficients and possibly growing first-order coefficients. We look for solutions which are summable to the $p$-th power with respect to the usual Lebesgue measure along with their first and second-order derivatives with respect to the spatial variable.