In this thesis, we study the regularity of optimal transport maps
and its applications to the semi-geostrophic system. The first two
chapters survey the known theory, in particular there is a
self-contained proof of Brenier' theorem on existence of optimal
transport maps and of Caffarelli's Theorem on Holder continuity of
optimal maps. In the third and fourth chapter we start
investigating Sobolev regularity of optimal transport maps, while
in Chapter 5 we show how the above mentioned results allows to
prove the existence of Eulerian solution to the semi-geostrophic
equation. In Chapter 6 we prove partial regularity of optimal maps
with respect to a generic cost functions (it is well known that in
this case global regularity can not be expected). More precisely we
show that if the target and source measure have smooth densities
the optimal map is always smooth outside a closed set of measure
zero.
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