Regularity of optimal transport on compact, locally nearly spherical, manifolds. (English) Zbl 1200.58025
Summary: Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold \(M\), we look for a smooth optimal transportation map \(G\), pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge-Ampère type satisfied by the potential function \(u\), such that \(G = \exp(\operatorname{grad} u)\). This approach boils down to proving an a priori upper bound on the Hessian of \(u\), which was done on the flat torus by the first author. The recent local \(C^{2}\) estimate of Ma-Trudinger-Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image \(G(m)\) of a generic point \(m \in M\), uniformly away from the cut-locus of \(m\); he checked a fourth-order inequality satisfied by the squared distance cost function, proving its so-called (strict) regularity. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in \(C^{2}\) norm specifying and proving a conjecture stated by Trudinger.
MSC:
| 58J90 | Applications of PDEs on manifolds |
| 58J05 | Elliptic equations on manifolds, general theory |
| 90B06 | Transportation, logistics and supply chain management |
| 35J96 | Monge-Ampère equations |
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