Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optimal transport maps
Guido De Philippis, Alessio Figalli
TL;DR
This paper proves strong second-order stability for solutions of the Monge-Ampere equation under certain conditions, leading to improved stability results for optimal transport maps.
Contribution
It establishes strong $W^{2,1}_{loc}$ convergence of Alexandrov solutions of the Monge-Ampere equation, enhancing understanding of stability in optimal transport.
Findings
Strong $W^{2,1}_{loc}$ convergence of solutions
Strong $W^{1,1}_{loc}$ stability of optimal transport maps
Convergence results hold under boundedness conditions on the right hand side
Abstract
The aim of this note is to show that Alexandrov solutions of the Monge-Ampere equation, with right hand side bounded away from zero and infinity, converge strongly in if their right hand side converge strongly in . As a corollary we deduce strong stability of optimal transport maps.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations