Mass transport problems obtained as limits of p-Laplacian type problems with spatial dependence

Abstract

We consider the following problem: given a bounded convex domain Ω⊂ℝN${\Omega \subset \mathbb {R}^N}$ we consider the limit as p → ∞ of solutions to -div(bp-p|Du|p-2Du)=f+-f-${- \operatorname{div} (b_{p}^{-p} |Du|^{p-2} Du)=f_+ - f_-}$ in Ω and bp-p|Du|p-2∂u∂η=0${ b_{p}^{-p} |Du|^{p-2} \frac{\partial u}{\partial \eta }=0}$ on ∂Ω${\partial \Omega }$. Under appropriate assumptions on the coefficients bp that in particular verify that limp→∞bp=b${ \lim _{p\rightarrow \infty } b_p = b }$ uniformly in Ω¯${\overline{\Omega }}$, we prove that there is a uniform limit of upj${u_{p_j}}$ (along a sequence pj→∞${p_j \rightarrow \infty }$) and that this limit is a Kantorovich potential for the optimal mass transport problem of f+${f_+}$ to f-${f_-}$ with cost c(x,y)${c(x,y)}$ given by the formula c(x,y)=infσ(0)=x,σ(1)=y∫σbds${c(x,y) = \inf _{\sigma (0)=x,\,\sigma (1)=y} \int _\sigma b\, ds}$.

Keywords

• mass transport
• monge–kantorovich problems
• p-laplacian equation
• 49j20
• 49j45
• 45g10