Local -minimizers versus local -minimizers of nonsmooth functionals

Abstract

We study not necessarily differentiable functionals of the form J ( u ) = 1 p ∫ Ω | ∇ u | p d x + 1 p ∫ Ω | u | p d x + ∫ Ω j 1 ( x , u ) d x + ∫ ∂ Ω j 2 ( x , γ u ) d σ with 1 < p < ∞ involving locally Lipschitz functions j 1 : Ω × R → R as well as j 2 : ∂ Ω × R → R . We prove that local C 1 ( Ω ¯ ) -minimizers of J must be local W 1 , p ( Ω ) -minimizers of J .