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]dx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {F}}(u) := \\int _\\Omega \\big [\ frac{1}{p} \\big (|Du|-1)^p_+ + f\\cdot u\\big ]\\mathrm {d}x\ onumber \\end{aligned}$$\\end{document}for p>1 in the vectorial case of mappings u:{mathbb {R}}^nsupset Omega rightarrow {mathbb {R}}^N with Nge 1. Assuming that f belongs to L^{n+sigma } for some sigma >0, we prove that {mathcal {H}}(Du) is continuous in Omega for any continuous function {mathcal {H}}:{mathbb {R}}^{Nn}rightarrow {mathbb {R}}^{Nn} vanishing on {xi in {mathbb {R}}^{Nn} : |xi |le 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 nge 2, to the vectorial case Nge 1.