Accepted Paper
(2008)
Authors:
Alessio Figalli - Grégoire Loeper
Journal: Calc. Var. Partial Differential Equations [MathSciNet]
Abstract:
We prove C1 regularity of c-convex weak Alexandrov solutions
of a Monge-Ampère type equation in dimension two, assuming only
a bound from above on the Monge-Ampère measure. The
Monge-Ampère equation involved arises in the optimal transport
problem. Our result holds true under a natural condition on the
cost function, namely non-negative cost-sectional curvature,
a condition introduced by Ma-Trudinger-Wang, that was shown to be necessary for C1 regularity. Such a
condition holds in particular for the case ``cost = distance
squared'' which leads to the usual Monge-Ampère equation detD2u = f. Our result is in some sense optimal, both for the
assumptions on the density (thanks to the regularity
counterexamples of X.J.Wang) and for the assumptions
on the cost-function (thanks to the results of the second author).
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[BibTeX Entry]
Available Files:
dim2Aw.pdf
abstract.tex (abstract.ps, abstract.pdf)
