The p-harmonic system with measure-valued right hand side

Abstract

For 2-12<p<n we prove existence of a distributional solution u of the p-harmonic system -div(|∇u|p-2∇u)=μinΩu=0on∂Ω, where Ω is an open subset of ℝn (bounded or unbounded), u : Ω → ℝm, and μ is an ℝm-valued Radon measure of finite mass. For the solution u we establish the Lorentz space estimate ‖Du‖Lq,∞+‖u‖Lq*,∞≤C‖μ‖M1p-1with q=nn-1(p-1) and q*=nn-p(p-1). The main step in the proof is to show that for suitable approximations the gradients Duk converge a.e. This is achieved by a choice of regularized test functions and a localization argument to compensate for the fact that in general u ∉ W1,p.