This book concerns with the regularity of optimal transport
maps and its applications to the semi-geostrophic system.
The first two chapters survey the known theory;
in particular a self-contained proof of Brenier’s theorem
on existence of optimal transport maps and of Caffarelli’s
theorem on Hölder continuity of optimal maps are included.
In the third and fourth chapters the Sobolev regularity
of optimal transport maps is involved, while in Chapter 5
the above-mentioned results lead to a proof
of the existence of an Eulerian solution to the
semi-geostrophic equation. Chapter 6 is about
partial regularity of optimal maps with respect to a generic
cost function. More precisely it is shown that if the target
and source measures have smooth densities then
the optimal map is always smooth outside a closed set
of measure zero.