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

We derive global gradient estimates for W^{1,p}_0(Omega)-weak solutions to quasilinear elliptic equations of the form div a(x, u,Du) = 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 Hoelder continuous in u. In the case when p > 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 Hoelder 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.