Gradient bounds for p-harmonic systems with vanishing Neumann (Dirichlet) data in a convex domain

摘要

Let Ω be a bounded convex domain in Euclidean n space, x ∈ ?Ω, and r > 0. Let ? = (?~1, ?~2,..., ?~N) be a weak solution to We show that sub solution type arguments for certain uniformly elliptic systems can be used to deduce that |??| is bounded in Ω ∩ B(x, r) with constants depending only on n, p, N, and r~n/|Ω ∩B(x,r)|. Our argument replaces an argument based on level sets in recent important work of Cianchi and Maz'ya (2014) [1,2], Geng and Shen (2010) [3], Maz'ya (2009) [4,5], involving similar problems. We also show that an analogous argument gives the same conclusion in the case of vanishing Dirichlet data.