Besov regularity for a class of singular or degenerate elliptic equations

TL;DR

This paper investigates the regularity of solutions to a class of singular or degenerate elliptic PDEs motivated by traffic flow models, establishing higher integrability and fractional differentiability results, especially for subquadratic growth cases.

Contribution

It introduces new regularity results for elliptic equations with subquadratic growth, a case previously neglected, including higher integrability and fractional differentiability of solutions.

Findings

Higher integrability of the gradient for solutions with data in Sobolev or Besov spaces.

Fractional differentiability of solutions in the subquadratic case.

Besov regularity results for cases with q ≥ 2.

Abstract

Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE $-\mathrm{div}\left((\vert\nabla u\vert-1)_{+}^{q-1}\frac{\nabla u}{\vert\nabla u\vert}\right)=f$, $\mathrm{in}\,\,\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $1<q<\infty$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. We assume that the datum $f$ belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of subquadratic growth, i.e. $1<q<2$, which has so far been neglected. In the latter case, we also prove the higher fractional differentiability of the solution to a variational problem, which is characterized by the above equation. For the sake of completeness, we finally give a Besov regularity result also in the case $q\geq2$.