Up-to-Boundary Pointwise Gradient Estimates for Very Singular Quasilinear Elliptic Equations with Mixed Data

Abstract

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 { - div ⁡ ( 𝐀 ⁢ ( x , D ⁢ u ) ) = g - div ⁡ f in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(\mathbf{A}(x,Du))&% \displaystyle=g-\operatorname{div}f&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is sufficiently flat in the sense of Reifenberg.