>>544


”Monge-Ampere Differential Equation
The solutions are given by a system of differential equations given by Iyanaga and Kawada (1980).”か
https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2018/Figalli-Citation.pdf
? Figalli short citation ?
For contributions to the theory of optimal transport and its applications in
partial differential equations, metric geometry and probability.

? Figalli long citation ?
Alessio Figalli has made multiple fundamental advances in the theory of
optimal transport, while also applying this theory in novel ways to other
areas of mathematics. Only a few of his numerous results in these areas are
described here.
Figalli’s joint work with De Philippis on regularity for the Monge-Amp`ere
equation is a groundbreaking result filling the gap between gradient estimates
discovered by Caffarelli and full Sobolev regularity of the second derivatives
of the convex solution of the Monge-Amp`ere equation with merely bounded
right-hand side. The result is almost optimal in view of existing counterexamples. It has direct implications on regularity of the optimal transport
maps, and on regularity to semigeostrophic equations.
Figalli initiated the study of the singular set of optimal transport maps and
obtained the first definite results in this direction: he showed that it has
null Lebesgue measure in full generality. He has also given significant contributions to the theory of obstacles problems, introducing new methods to
analyze the structure of the free boundary.

つづく