Abstract
We indicate how recent work of Figalli–Kim–McCann and Vetois can be used to improve previous results of Trudinger and Wang on classical solvability of the second boundary value problem for Monge–Ampere type partial differential equations arising in optimal transportation together with the global regularity of the associated optimal mappings.
Highlights
In this short note we use recent work of Figalli–Kim–McCann [4] and Vetois [24] on continuous differentiability and strict convexity of potentials in optimal transportation to improve previous results on global regularity of optimal mappings in [20]
A weaker interpretation of the boundary condition (1.5) arises through optimal transportation, in which case Caffarelli [1] proved that the convexity of the target ∗ suffices for local smoothness of solutions
Corollary 1.2 Under the hypotheses of Theorem 1, there exists a unique diffeomorphism T = Y (·, Du), ∈ [C2( )]n maximizing the functional (1.8), where u is an elliptic solution of the second boundary value problem (1.3), (1.5)
Summary
In this short note we use recent work of Figalli–Kim–McCann [4] and Vetois [24] on continuous differentiability and strict convexity of potentials in optimal transportation to improve previous results on global regularity of optimal mappings in [20]. Keywords Optimal transportation · Monge–Ampère equations · Second boundary value problem
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