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

Abstract

Abstract. We consider the following problem: given a bounded convex domain Ω ⊂ ℝ N ${\Omega \subset \mathbb {R}^N}$ we consider the limit as p → ∞ of solutions to - div ( b p - p | D u | p - 2 D u ) = f + - f - ${- \operatorname{div} (b_{p}^{-p} |Du|^{p-2} Du)=f_+ - f_-}$ in Ω and b p - p | D u | 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 lim p → ∞ b p = b ${ \lim _{p\rightarrow \infty } b_p = b }$ uniformly in Ω ¯ ${\overline{\Omega }}$ , we prove that there is a uniform limit of u p j ${u_{p_j}}$ (along a sequence p j → ∞ ${p_j \rightarrow \infty }$ ) 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 ) = inf σ ( 0 ) = x , σ ( 1 ) = y ∫ σ b d s ${c(x,y) = \inf _{\sigma (0)=x,\,\sigma (1)=y} \int _\sigma b\, ds}$ .