On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds

Carlier, Guillaume; Brasco, Lorenzo

Carlier, Guillaume; Brasco, Lorenzo (2013), On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds, Advances in Calculus of Variations, 7, 3, p. 379–407. http://dx.doi.org/10.1515/acv-2013-0007

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00722615

Date

2013

Journal name

Advances in Calculus of Variations

Volume

Number

Publisher

De Gruyter

Pages

379–407

Abstract (EN)

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 $\nabla u\in L^r_{loc}$ for every $r\ge 1$.

Subjects / Keywords

Traffic congestion; Anisotropic problems; Degenerate elliptic equations