Higher regularity in congested traffic dynamics

In this paper, we consider minimizers of integral functionals of the type F(u) := integral(Omega) [1/p(vertical bar Du vertical bar - 1)(+)(p) + f . u]dx for p > 1 in the vectorial case ofmappings u : R-n superset of Omega R-N with N >= 1. Assuming that f belongs to Ln+sigma for some sigma > 0, we prove that H(Du) is continuous in Omega for any continuous function H: R-Nn -> R-Nn vanishing on {xi epsilon R-Nn : vertical bar xi vertical bar <= 1}. This extends previous results of Santambrogio and Vespri (Nonlinear Anal 73:3832-3841, 2010) when n = 2, and Colombo and Figalli (J Math Pures Appl (9) 101(1):94-117, 2014) for n >= 2, to the vectorial case N >= 1.