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On the second boundary value problem for Monge-Ampère type equations and optimal transportation. (English) Zbl 1182.35134

The paper is concerned with the existence of globally smooth solutions for the second boundary value problem for the Monge-Ampère equation
\[ \text{det}\{D^2u -A(\cdot,u,Du\}=B(\cdot,u,Du) \]
and the application to regularity of potentials in optimal transportation. In particular, there are determined conditions on costs and domains to ensure that optimal mappings are smooth diffeomorphisms. The authors resolve, in the context of global regularity, the fundamental problem of regularity for more general costs than the quadratic costs. The approach is through the derivation of global estimates for second derivatives of solutions.

MSC:

35J96 Monge-Ampère equations
35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
49K20 Optimality conditions for problems involving partial differential equations
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