Global Sobolev regularity for general elliptic equations of <Emphasis Type="Italic">p</Emphasis>-Laplacian type

We derive global gradient estimates for \(W^{1,p}_0(\Omega )\)-weak solutions to quasilinear elliptic equations of the form

$$\begin{aligned} \mathrm {div\,}\mathbf {a}(x,u,Du)=\mathrm {div\,}(|F|^{p-2}F) \end{aligned}$$

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 merely continuous in u. Our result highly improves the known regularity results available in the literature. Actually, we are able not only to weaken the Lipschitz continuity with respect to u of the nonlinearity to only uniform continuity, but we also find a very lower level of geometric assumption on the boundary of the domain to ensure a global character of the gradient estimates obtained.