Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth

TL;DR

This paper proves that weak solutions to certain quasilinear elliptic and parabolic equations are globally regular under specific growth and coefficient conditions, advancing understanding of solution smoothness in complex domains.

Contribution

It establishes global regularity results for weak solutions with BMO coefficients and controlled growth, extending previous regularity theories to broader classes of equations.

Findings

Weak solutions are globally regular under specified conditions.

BMO coefficients with small mean oscillation are sufficient for regularity.

The results apply to equations on Lipschitz domains.

Abstract

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.