Higher regularity in congested traffic dynamics

Abstract

AbstractIn this paper, we consider minimizers of integral functionals of the type $$\begin{aligned} {\mathcal {F}}(u) := \int _\Omega \big [\tfrac{1}{p} \big (|Du|-1)^p_+ + f\cdot u\big ]\mathrm {d}x\nonumber \end{aligned}$$ F ( u ) : = ∫ Ω [ 1 p ( | D u | - 1 ) + p + f · u ] d x for $$p>1$$ p > 1 in the vectorial case of mappings $$u:{\mathbb {R}}^n\supset \Omega \rightarrow {\mathbb {R}}^N$$ u : R n ⊃ Ω → R N with $$N\ge 1$$ N ≥ 1 . Assuming that f belongs to $$L^{n+\sigma }$$ L n + σ for some $$\sigma >0$$ σ > 0 , we prove that $${\mathcal {H}}(Du)$$ H ( D u ) is continuous in $$\Omega $$ Ω for any continuous function $${\mathcal {H}}:{\mathbb {R}}^{Nn}\rightarrow {\mathbb {R}}^{Nn}$$ H : R Nn → R Nn vanishing on $$\{\xi \in {\mathbb {R}}^{Nn} : |\xi |\le 1\}$$ { ξ ∈ R Nn : | ξ | ≤ 1 } . This extends previous results of Santambrogio and Vespri (Nonlinear Anal 73:3832–3841, 2010) when $$n=2$$ n = 2 , and Colombo and Figalli (J Math Pures Appl (9) 101(1):94–117, 2014) for $$n\ge 2$$ n ≥ 2 , to the vectorial case $$N\ge 1$$ N ≥ 1 .