Lipschitz regularity for constrained local minimizers of convex variational integrals with a wide range of anisotropy

We establish interior gradient bounds for functions ${u in W^1_{1, {rm loc}} (Omega)}$ which locally minimize the variational integral ${J [u, Omega] = int_Omega h left( |nabla u| right) dx}$ under the side condition ${u ge Psi}$ a.e. on Ω with obstacle ${Psi}$ being locally Lipschitz. Here h denotes a rather general N-function allowing (p, q)-ellipticity with arbitrary exponents 1 < p ≤ q < ∞. Our arguments are based on ideas developed in Bildhauer et al. (Z Anal Anw 20:959–985, 2001) combined with techniques originating in Fuchs (2011).