Gradient bounds for p-harmonic systems with vanishing neumann data in a convex domain

Let $\text{ti}\text{Om}$ be a bounded convex domain in Euclidean $n$ space, $ \hat x \in \ar \ti \Om, $and$ r > 0. $Let$ \ti u = (\ti u^1, \ti u^2, \dots, \ti u^N) $ be a weak solution to $$\mathrm{\nabla }\cdot \left(\mathrm{\mid }\mathrm{\nabla }\text{ti}u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }\text{ti}u\right)=0in\text{ti}\text{Om}\cap B(\widehat{x},4r)with\mathrm{\mid }\mathrm{\nabla }\text{ti}u{\mathrm{\mid }}^{p-2}\text{ti}{u}_{\nu }=0on\text{ar}\text{ti}\text{Om}\cap B(\widehat{x},4r)\mathrm{.}$$ We show that sub solution type arguments for certain uniformly elliptic systems can be used to deduce that $\mathrm{\mid }\mathrm{\nabla }\text{ti}u\mathrm{\mid }$ is bounded in $ \ti \Om \cap B (\hat x, r)$withconstantsdependingonlyon$ n, p, N. $and$ \frac{r^n}{| \ti \Om \cap B (\hat x, r) |}.$ Our argument replaces an argument based on level sets in recent important work of [CM], [CM1], [GS], [M], [M1], involving similar problems.