We prove the existence of solutions to the Monge problem with an absolutely continuous
initial measure by solving a secondary variational problem with any strictly convex function,
the so-called secondary variational method. The cost function is given by an arbitrary
norm on
.
In addition, if a norm satisfies the uniform smoothness and convexity estimates, and
two measures are absolutely continuous, then for the Monge problem with such a
norm cost function, we can find a same optimal transport map via the secondary
variational method even with different strictly convex functions (the classical Monge
problem is a special case). This optimal transport map is just the one which uniquely
satisfies a monotone condition. Finally, we construct an example with the
norm
cost function, which is not a strictly convex norm, to show that one can get different
optimal transport maps by solving secondary variational problems with different
strictly convex functions. In view of this example, for the Monge problem between
two absolutely continuous measures, if a norm cost function does not satisfy the
uniform smoothness and convexity estimates, there can be no uniqueness of
optimal transport maps obtained via the secondary variational method.
Keywords
Monge problem, optimal transport map, secondary variational
problem
