Some Hints and Discussions about the Regularity Problem for the Case \(c(x,y)=|x-y|\) and its Approximation
speaker: Filippo Santambrogio (Université Claude Bernard Lyon 1)abstract:
It is well known from the 2005 paper by Ma-Trudinger-Wang that some transport costs enjoy good properties for regularity results, and that other do not. Among the power costs \(|x-y|^p\) used for Wasserstein distances, only \(p=2\) satisfies the MTW inequality which guarantees that one can get regularity results. Yet, for every \(\varepsilon>0\), the cost \(c_\varepsilon(x,y)=\sqrt{\varepsilon^2+|x-y|^2}\) satisfies MTW assumptions and \(\lim_{\varepsilon\to 0}c_\varepsilon (x,y)=|x-y|\). This naturally raises the question of the regularity for this limit cost, which was the original one proposed by Monge. In this particular case, rather than the regularityh of the potential, we are more interested in the regularity for the optimal transport itself, and in particular for the one which is monotone on transport rays. This would also have consequences on the regulairy of the so-called trasnport density, a measure accounting for the local amount of traffic induced at any point by the optimal transport, peculiar of the case \(c(x,y)=|x-y|\). Some results by Fragalà, Gelli and Pratelli exist in two dimensions and under strong assumptions on the supports of the two measures, and in general one cannot expect as much regularity as in the case \(\varepsilon>0\).
After discussing the problem, the results that one can expect, the difficulties and the key ingredients, I will show some estimates which are uniform as \(\varepsilon\to 0\).
Unfortunately, this is far from giving the desired results...
timetable:
Tue 6 Nov, 10:40 - 11:30, Aula Dini
<< Go back