Nonlinear elliptic problems with $p$-structure under mixed boundary value conditions in polyhedral domains

Abstract

Nonlinear elliptic systems with $p$-structure for $p \! \geq \! 2$, such as \[ -\mbox{div}\left((1+|\nabla u|^{p-2})\nabla u\right) =f(x)+\sum_{i=1}^n\partial_if_i(x) ,\] are considered under mixed boundary value conditions on nonsmooth domains. Regularity results in fractional-order Sobolev spaces are proven, e.g., $u\in W^{r,p}(\Omega)$ for all $r <1+\frac{1}{p}$ and $|\nabla u|^p\in W^{s,1}(\Omega)$ for some $s>1$.