The polar cone of the set of monotone maps
Authors:
Fabio Cavalletti and Michael Westdickenberg
Journal:
Proc. Amer. Math. Soc. 143 (2015), 781-787
MSC (2010):
Primary 49Q20
DOI:
https://doi.org/10.1090/S0002-9939-2014-12332-X
Published electronically:
October 15, 2014
MathSciNet review:
3283664
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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.
- [1]
Giovanni Alberti and Luigi Ambrosio, A geometrical approach to monotone functions in
, Math. Z. 230 (1999), no. 2, 259-316. MR 1676726 (2000d:49033), https://doi.org/10.1007/PL00004691
- [2] 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 (2009h:74055), https://doi.org/10.1142/S0218202508003133
- [3] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562 (50 #1060)
- [4] 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 (2009g:49004)
- [5] 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 (86k:28001)
- [6] 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 (99i:49039), https://doi.org/10.1016/S0764-4442(98)80397-2
- [7] 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 (2010m:35306), https://doi.org/10.1137/090750809
- [8] John Riedl, Partially ordered locally convex vector spaces and extensions of positive continuous linear mappings, Math. Ann. 157 (1964), 95-124. MR 0169033 (29 #6288)
- [9] 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 (2011k:35137), https://doi.org/10.1142/S0219891610002244
- [10] 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 (52 #9014)
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Additional Information
Fabio Cavalletti
Affiliation:
Lehrstuhl für Mathematik (Analysis), RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany
Email:
cavalletti@instmath.rwth-aachen.de
Michael Westdickenberg
Affiliation:
Lehrstuhl für Mathematik (Analysis), RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany
Email:
mwest@instmath.rwth-aachen.de
DOI:
https://doi.org/10.1090/S0002-9939-2014-12332-X
Keywords:
Optimal transport,
polar cone
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
Article copyright:
© Copyright 2014
American Mathematical Society
