Global regularity for solutions to Monge-Ampère type equations
The optimal transportation problem was formulated by Monge in 1781: given two domains $\Omega, \Omega^* \subset \R^n$ and two mass distributions $f\in L^1(\Omega)$ and $g\in L^1(\Omega^*)$ of the same mass, find the optimal volume-preserving map $T$ between them, where optimality is measured against a cost functional $$ \mathcal C(s)=\int_\Omega f(x) c(x,s(x)). $$ %One views the first set $\Omega$ as being filled with mass, %and $c(x,y)$ as being the cost for transporting per unit mass from $x...[Show more]
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