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

Abstract

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.