Higher regularity in congested traffic dynamics

In this paper, we consider minimizers of integral functionals of the type F ( u ) : = ∫ Ω [ 1 p ( | D u | - 1 ) + p + f · u ] d x for p > 1 in the vectorial case of mappings u : R n ⊃ Ω → R N with N ≥ 1 . Assuming that f belongs to L n + σ for some σ > 0 , we prove that H ( D u ) is continuous in Ω for any continuous function H : R Nn → R Nn vanishing on { ξ ∈ R Nn : | ξ | ≤ 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 .