# Bienvenue - Laboratoire Jacques-Louis Lions

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### Chiffres clefs

#### 217 personnes travaillent au LJLL

##### 83 personnels permanents

47 enseignants chercheurs

13 chercheurs CNRS

9 chercheurs INRIA

2 chercheurs CEREMA

12 ingénieurs, techniciens et personnels administratifs

##### 134 personnels non permanents

85 doctorants

16 post-doc et ATER

5 chaires et délégations

12 émérites et collaborateurs bénévoles

16 visiteurs

Chiffres janvier 2014

### GT CalVa R MCCANN

Lundi 9 janvier 2017

Robert MC CANN (University of Toronto Bahen Centre)

Free Discontinuities in Optimal Transport

Abstract : Optimal maps in $R^n$ to disconnected targets necessarily contain discontinuities (i.e. tears). But how smooth are these tears ? When the target components are suitably separated by hyperplanes, non-smooth versions of the implicit function theorem can be developed which show the tears are hypersurfaces given as differences of convex functions --- DC for short. If in addition the targets are convex the tears are actually $C^1,\alpha$. Similarly, under suitable affine independence assumptions, singularities of multiplicity $k$ lie on DC rectifiable submanifolds of dimension $n+1-k$. These are stable with respect to $W_\infty$ perturbations of the target measure. Moreover, there is at most one singularity of multiplicity $n$. This represents joint work with Jun Kitagawa.