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

Abstract

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: ∂ x ( | u x | - δ 1 ) + q - 1 u x | u x | + ∂ y ( | u y | - δ 2 ) + q - 1 u y | u y | = f , $ \partial _x \biggl [(|u_{x}|-\delta _1)_+^{q-1}\, \frac{u_{x}}{|u_{x}|}\biggr ]+\partial _y \biggl [(|u_{y}|-\delta _2)_+^{q-1}\, \frac{u_{y}}{|u_{y}|}\biggr ]=f, $ for 2 ≤ q < ∞ ${2\le q<\infty }$ and some non-negative parameters δ 1 , δ 2 ${\delta _1,\delta _2}$ . Here ( · ) + ${(\,\cdot \,)_+}$ stands for the positive part. We prove that if f ∈ L loc ∞ ${f\in L^\infty _{\rm loc}}$ , then ∇ u ∈ L loc r ${\nabla u\in L^r_{\rm loc}}$ for every r ≥ 1.

Optimal regions for congested transport

“…An optimizer u then gives the price system which maximizes the profit of the company. When you take into account a congested transport between sources (here called f + and f − ), the total mass plays an important role: as observed in [6], in the case of a small mass, hence of a large Lagrange multiplier k, the congestion effects are negligible, so one can expect in this case a distribution of the low-congestion region around the source distribution. On the contrary, for a large mass, hence for a small Lagrange multiplier k, we may expect a distribution of the low-congestion region also between the sources f + and f − .…”

“…In the present paper we consider a very simplified model in which the densities of residents and of working places are known, represented by two probability measures f + and f − . The congestion effects in mass transportation theory has been deeply studied in the literature; we refer for instance to [15,6] and references therein. Denoting by f the difference f = f + − f − and by σ the traffic flux, the model, in the stationary regime, reduces to a minimization problem of the form min Ω H(σ) dx : − div σ = f in Ω, σ · n = 0 on ∂Ω .…”

Regularity properties of viscosity solutions for fully nonlinear equations on the model of the anisotropic p →-Laplacian

“…Among the recent regularity results obtained, let us cite the paper of Brasco and Carlier [7], which proves that for the widely degenerate anisotropic equation, arising in congested optimal transport…”

A weak comparison principle for solutions of very degenerate elliptic equations

“…Our interest in this kind of equations comes from recent studies in traffic congestion problems (see [2] and [3]), complex-valued solutions of the eikonal equation (see [13]- [16]) and in variational problems which are relaxations of non-convex ones (see for instance [4] and [10]).…”