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

Lorenzo Brasco (1) , Guillaume Carlier (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 $r\ge 1$.

Mots clés: Degenerate elliptic equations; Anisotropic problems; Traffic congestion

Domaines: Equations aux dérivées partielles [math.AP]

Lorenzo Brasco, Guillaume Carlier. On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds. Advances in Calculus of Variation, 2014, 7 (3), pp.379-407. ⟨10.1515/acv-2013-0007⟩. ⟨hal-00722615⟩