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

Let Ω̃ be a bounded convex domain in Euclidean nn space, xˆ∈∂Ω̃, and r>0r>0. Let ũ=(ũ1,ũ2,…,ũN) be a weak solution to ∇⋅(|∇ũ|p−2∇ũ)=0in  Ω̃∩B(xˆ,4r)with  |∇ũ|p−2ũν=0on  ∂Ω̃∩B(xˆ,4r). We show that sub solution type arguments for certain uniformly elliptic systems can be used to deduce that |∇ũ| is bounded in Ω̃∩B(xˆ,r) with constants depending only on n,p,Nn,p,N, and rn|Ω̃∩B(xˆ,r)|. Our argument replaces an argument based on level sets in recent important work of Cianchi and Maz’ya (2014)  [1] and [2], Geng and Shen (2010)  [3], Maz’ya (2009)  [4] and [5], involving similar problems. We also show that an analogous argument gives the same conclusion in the case of vanishing Dirichlet data.