The polar cone of the set of monotone maps
HTML articles powered by AMS MathViewer
- by Fabio Cavalletti and Michael Westdickenberg PDF
- Proc. Amer. Math. Soc. 143 (2015), 781-787 Request permission
Abstract:
We prove that every element of the polar cone to the closed convex cone of monotone transport maps can be represented as the divergence of a measure field taking values in the positive definite matrices.References
- Giovanni Alberti and Luigi Ambrosio, A geometrical approach to monotone functions in $\textbf {R}^n$, Math. Z. 230 (1999), no. 2, 259–316. MR 1676726, DOI 10.1007/PL00004691
- Guy Bouchitté, Wilfrid Gangbo, and Pierre Seppecher, Michell trusses and lines of principal action, Math. Models Methods Appl. Sci. 18 (2008), no. 9, 1571–1603. MR 2446402, DOI 10.1142/S0218202508003133
- H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). MR 0348562
- G. Carlier and T. Lachand-Robert, Representation of the polar cone of convex functions and applications, J. Convex Anal. 15 (2008), no. 3, 535–546. MR 2431410
- Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633
- Pierre-Louis Lions, Identification du cône dual des fonctions convexes et applications, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 12, 1385–1390 (French, with English and French summaries). MR 1649179, DOI 10.1016/S0764-4442(98)80397-2
- Luca Natile and Giuseppe Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal. 41 (2009), no. 4, 1340–1365. MR 2540269, DOI 10.1137/090750809
- John Riedl, Partially ordered locally convex vector spaces and extensions of positive continuous linear mappings, Math. Ann. 157 (1964), 95–124. MR 169033, DOI 10.1007/BF01362669
- Michael Westdickenberg, Projections onto the cone of optimal transport maps and compressible fluid flows, J. Hyperbolic Differ. Equ. 7 (2010), no. 4, 605–649. MR 2746202, DOI 10.1142/S0219891610002244
- Eduardo H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 237–341. MR 0388177
Additional Information
- Fabio Cavalletti
- Affiliation: Lehrstuhl für Mathematik (Analysis), RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany
- MR Author ID: 956139
- Email: cavalletti@instmath.rwth-aachen.de
- Michael Westdickenberg
- Affiliation: Lehrstuhl für Mathematik (Analysis), RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany
- MR Author ID: 654309
- Email: mwest@instmath.rwth-aachen.de
- Received by editor(s): May 10, 2013
- Received by editor(s) in revised form: June 3, 2013
- Published electronically: October 15, 2014
- Communicated by: Walter Craig
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 781-787
- MSC (2010): Primary 49Q20
- DOI: https://doi.org/10.1090/S0002-9939-2014-12332-X
- MathSciNet review: 3283664