Local C1((Ω)over-bar)-minimizers versus local W1,p(Ω)-minimizers of nonsmooth functionals

We study not necessarily differentiable functionals of the form J(u) = 1/p integral(Omega)vertical bar del u vertical bar(p)dx + 1/p integral(Omega)vertical bar u vertical bar(p)dx + integral(Omega)j(1)(x, u)dx + integral(partial derivative Omega)j(2)(x, gamma u)d sigma with 1 < p < infinity involving locally Lipschitz functions j(1) : Omega x R -> R as well as j(2) : partial derivative Omega x R -> R. We prove that local C-1((Omega) over bar)-minimizers of J must be local W-1,W-p(Omega)-minimizers of J. (C) 2010 Elsevier Ltd. All rights reserved.