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

We study not necessarily differentiable functionals of the form J(u)=1p∫Ω|∇u|pdx+1p∫Ω|u|pdx+∫Ωj1(x,u)dx+∫∂Ωj2(x,γu)dσ with 1