# Global Sobolev regularity for general elliptic equations of $p$-Laplacian type

## TL;DR

This paper establishes improved global gradient estimates for solutions to a class of quasilinear elliptic equations with minimal boundary regularity assumptions, extending regularity results to less smooth nonlinearities and domains.

## Contribution

It introduces new global Sobolev regularity results for elliptic equations with small-BMO nonlinearities on Reifenberg flat domains, relaxing previous regularity and geometric assumptions.

## Key findings

- Global gradient estimates for solutions are derived.
- Regularity requirements on the nonlinearity are weakened from Lipschitz to Hölder.
- The geometric assumptions on the domain boundary are significantly reduced.

## Abstract

We derive global gradient estimates for $W^{1,p}_0(\Omega)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and H\"older continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonlinearity in $u$ from Lipschitz continuity to H\"older one, but we also find a very lower level of geometric assumptions on the boundary of the domain to ensure global character of the obtained gradient estimates.