no. 3
Local semiconvexity of Kantorovich potentials on non-compact manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 648-653.

We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n - 1-dimensional rectifiable sets.

DOI : 10.1051/cocv/2010011
Classification : 49Q20, 35J96
Mots clés : Kantorovich potential, optimal transport, regularity 
@article{COCV_2011__17_3_648_0,
     author = {Figalli, Alessio and Gigli, Nicola},
     title = {Local semiconvexity of {Kantorovich} potentials on non-compact manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {648--653},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {3},
     year = {2011},
     doi = {10.1051/cocv/2010011},
     mrnumber = {2826973},
     zbl = {1228.49047},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2010011/}
}
TY  - JOUR
AU  - Figalli, Alessio
AU  - Gigli, Nicola
TI  - Local semiconvexity of Kantorovich potentials on non-compact manifolds
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2011
SP  - 648
EP  - 653
VL  - 17
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2010011/
DO  - 10.1051/cocv/2010011
LA  - en
ID  - COCV_2011__17_3_648_0
ER  - 
%0 Journal Article
%A Figalli, Alessio
%A Gigli, Nicola
%T Local semiconvexity of Kantorovich potentials on non-compact manifolds
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 648-653
%V 17
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2010011/
%R 10.1051/cocv/2010011
%G en
%F COCV_2011__17_3_648_0
Figalli, Alessio; Gigli, Nicola. Local semiconvexity of Kantorovich potentials on non-compact manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 648-653. doi : 10.1051/cocv/2010011. http://www.numdam.org/articles/10.1051/cocv/2010011/

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). | MR | Zbl

[2] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in spaces of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005). | MR | Zbl

[3] D. Cordero-Erasquin, R.J. Mccann and M. Schmuckenschlager, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146 (2001) 219-257. | MR | Zbl

[4] A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. (to appear). | MR | Zbl

[5] A. Figalli, Existence, uniqueness, and regularity of optimal transport maps. SIAM J. Math. Anal. 39 (2007) 126-137. | MR | Zbl

[6] W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | MR | Zbl

[7] N. Gigli, Second order analysis on (𝒫 2 (M),W 2 ) . Memoirs of the AMS (to appear), available at http://cvgmt.sns.it/cgi/get.cgi/papers/gig09/. | Zbl

[8] R.J. Mccann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608. | MR | Zbl

[9] C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin-New York (2009). | MR | Zbl

Cité par Sources :