Higher regularity in congested traffic dynamics

Abstract

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