## Abstract

Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, exhibiting a wide range of degeneracy. The model case we have in mind is the following: $\partial_x \left[(|u_{x}|-\delta_1)_+^{q-1}\, \frac{u_{x}}{|u_{x}|}\right]+\partial_y \left[(|u_{y}|-\delta_2)_+^{q-1}\, \frac{u_{y}}{|u_{y}|}\right]=f,$ for $2\le q<\infty$ and some non negative parameters $\delta_1,\delta_2$. Here $(\,\cdot\,)_+$ stands for the positive part. We prove that if $f\in L^\infty_{loc}$, then $\ abla u\in L^r_{loc}$ for every $r\ge 1$.

## Original document

The different versions of the original document can be found in:

https://basepub.dauphine.fr/handle/123456789/9874,
https://hal.archives-ouvertes.fr/hal-00722615/document,
https://hal.archives-ouvertes.fr/hal-00722615/document,
https://hal.archives-ouvertes.fr/hal-00722615/file/bracar0712.pdf

### Document information

Published on 01/01/2014

Volume 2014, 2014
DOI: 10.1515/acv-2013-0007
Licence: Other

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