Up-to-boundary pointwise gradient estimates for very singular quasilinear elliptic equations with mixed data

Abstract: This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data: \begin{cases} -\operatorname{div}(A(x,D u))=g-\operatorname{div} f \quad & \mathrm{in} \quad \Omega \ u= 0 \quad & \text{on} \ \partial \Omega, \end{cases} where $\Omega \subset \mathbb{R}^n$ is sufficiently flat in the sense of Reifenberg.