Scipediacontent: Scipediacontent moved page Draft Content 363988872 to Carlier Brasco 2014a

2021-02-16T10:16:10Z

Scipediacontent moved page Draft Content 363988872 to Carlier Brasco 2014a

Scipediacontent: Created page with " == Abstract == Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic ell..."

2021-02-16T10:16:06Z

Created page with " == Abstract == Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic ell..."

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== 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:

* [http://dx.doi.org/10.1515/acv-2013-0007 http://dx.doi.org/10.1515/acv-2013-0007] under the license http://hal.archives-ouvertes.fr/licences/copyright/

* [http://hal.inria.fr/docs/00/72/26/15/PDF/bracar0712.pdf http://hal.inria.fr/docs/00/72/26/15/PDF/bracar0712.pdf]

* [https://hal.archives-ouvertes.fr/hal-00722615 https://hal.archives-ouvertes.fr/hal-00722615],
: [https://basepub.dauphine.fr/handle/123456789/9874 https://basepub.dauphine.fr/handle/123456789/9874],
: [https://www.degruyter.com/view/j/acv.ahead-of-print/acv-2013-0007/acv-2013-0007.xml https://www.degruyter.com/view/j/acv.ahead-of-print/acv-2013-0007/acv-2013-0007.xml],
: [https://hal.archives-ouvertes.fr/hal-00722615/document https://hal.archives-ouvertes.fr/hal-00722615/document],
: [https://academic.microsoft.com/#/detail/2052758893 https://academic.microsoft.com/#/detail/2052758893]

* [https://hal.archives-ouvertes.fr/hal-00722615 https://hal.archives-ouvertes.fr/hal-00722615],
: [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 https://hal.archives-ouvertes.fr/hal-00722615/file/bracar0712.pdf]

* [https://www.degruyter.com/view/j/acv.2014.7.issue-3/acv-2013-0007/acv-2013-0007.pdf https://www.degruyter.com/view/j/acv.2014.7.issue-3/acv-2013-0007/acv-2013-0007.pdf],
: [http://dx.doi.org/10.1515/acv-2013-0007 http://dx.doi.org/10.1515/acv-2013-0007] under the license http://hal.archives-ouvertes.fr/licences/copyright/

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Scipediacontent: Scipediacontent moved page Draft Content 363988872 to Carlier Brasco 2014a

Scipediacontent: Created page with " == Abstract == Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic ell..."