Elliptic equations with measurable nonlinearities in nonsmooth domains (Q900848)

The paper deals with regularity of the weak solutions to the Dirichlet problem \[ \begin{cases} \operatorname{div} a(x,Du)= \operatorname{div} F \quad & \text{in } \Omega,\\ u=0 & \text{on } \partial\Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^n\) is a bounded and Reifenberg flat domain. The nonlinear term \(a(x,Du)\) is uniformly elliptic, it grows at most linearly with respect to \(Du\), and is allowed to be only measurable in one spatial direction while a small BMO condition is imposed with respect to the remaining \(n-1\) variables. The authors obtain global \(W^{1,p}\)-estimates for the weak solutions by proving that \[ \int_\Omega |Du|^p\;dx \leq C \int_\Omega |F|^p\;dx \] for any \(p\geq 2\).