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

Let $\ti \Om$ 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 $$\nabla \cdot \left (|\nabla \ti u |^{p-2} \nabla \ti u \right) = 0 \mbox{in} \ti \Om \cap B (\hat x, 4 r) \mbox{with} |\nabla \ti u|^{p-2} \, \ti u_\nu = 0 \mbox{on} \ar \ti \Om \cap B (\hat x, 4 r).$$ We show that sub solution type arguments for certain uniformly elliptic systems can be used to deduce that $| \nabla \ti u |$ is bounded in $\ti \Om \cap B (\hat x, r)$ with constants depending only on $n, p, N.$ 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.