Higher regularity in congested traffic dynamics

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