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SUMMARY:A variational approach to the regularity theory for optimal transp
ortation
DTSTART;VALUE=DATE-TIME:20210608T120000Z
DTEND;VALUE=DATE-TIME:20210608T130000Z
DTSTAMP;VALUE=DATE-TIME:20211022T233823Z
UID:indico-event-6615@indico.math.cnrs.fr
DESCRIPTION:The optimal transportation of one measure into another\, leadi
ng to the notion of their Wasserstein distance\, is a problem in the calcu
lus of variations with a wide range of applications. The regularity theory
for the optimal map is subtle and was pioneered by Caffarelli. That appro
ach relies on the fact that the Euler-Lagrange equation of this variationa
l problem is given by the Monge-AmpĂ¨re equation. The latter is a prime ex
ample of a fully nonlinear (degenerate) elliptic equation\, amenable to co
mparison principle arguments.\n\nWe present a purely variational approach
to the regularity theory for optimal transportation\, introduced with M. G
oldman. Following De Giorgi's philosophy for the regularity theory of mini
mal surfaces\, it is based on the approximation of the displacement by a h
armonic gradient through the construction of a variational competitor\, wh
ich leads to a "one-step improvement lemma"\, and feeds into a Campanato i
teration on the $C^{1\,\\alpha}$-level for the optimal map\, capitalizing
on affine invariance.\n\nOn the one hand\, this allows to reprove the $\\e
psilon$-regularity result (Figalli-Kim\, De Philippis-Figalli) bypassing C
affarelli's celebrated theory. This also extends to boundary regularity (C
hen-Figalli)\, which is joint work with T. Miura\, and to general cost fun
ctions\, which is joint work with M. Prodhomme and T. Ried\, based on the
notion of almost minimality.\n\nOn the other hand\, due to its robustness\
, it can be used as a large-scale regularity theory for the problem of mat
ching the Lebesgue measure to the Poisson measure in the thermodynamic lim
it. This is joint work with M. Goldman and M. Huesmann.\n\nhttps://indico.
math.cnrs.fr/event/6615/
LOCATION:ICJ Fokko
URL:https://indico.math.cnrs.fr/event/6615/
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