Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data

We study global regularity for solutions of quasilinear elliptic equations of the form $÷\text{A}(x,u,\mathrm{\nabla }u)=÷\text{F}$ in rough domains $\mathrm{\Omega }$ in ${\mathbb{R}}^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\text{A}$ is assumed to be continuous in $u$, and its growth in $\mathrm{\nabla }u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $\mathrm{\Omega }$, a small BMO condition in $x$ for $\text{A}$, and an optimal condition for the Dirichlet boundary data.