On divergence form second-order PDEs with growing coefficients in $W^{1}_{p}$ spaces without weights

TL;DR

This paper studies second-order divergence form PDEs with growing coefficients, establishing existence and regularity of solutions in unweighted Sobolev spaces without weights, under minimal regularity assumptions on coefficients.

Contribution

It extends the theory of divergence form PDEs to include linearly growing coefficients in unweighted Sobolev spaces, without requiring weighted spaces or strong regularity.

Findings

Existence of solutions in $W^{1}_{p}$ spaces without weights.

Regularity results for PDEs with linearly growing coefficients.

Applicable to uniformly parabolic and elliptic PDEs with VMO coefficients.

Abstract

We consider second-order divergence form uniformly parabolic and elliptic PDEs with bounded and $VMO_{x}$ leading coefficients and possibly linearly growing lower-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 derivatives with respect to the spatial variables.