Skip to main content
SpringerLink
Log in

A Note on Some Poincaré Inequalities on Convex Sets by Optimal Transport Methods

  • Conference paper
  • First Online:
Geometric Properties for Parabolic and Elliptic PDE's (GPPEPDEs 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 176))

  • 993 Accesses

Abstract

We show that a class of Poincaré-Wirtinger inequalities on bounded convex sets can be obtained by means of the dynamical formulation of Optimal Transport. This is a consequence of a more general result valid for convex sets, possibly unbounded.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Acosta, G., Durán, R.G.: An optimal Poincaré inequality in \(L^1\) for convex domains. Proc. Am. Math. Soc. 132, 195–202 (2004)

    Article  MATH  Google Scholar 

  2. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measure. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)

    MATH  Google Scholar 

  3. Agueh, M., Ghoussoub, N., Kang, X.: Geometric inequalities via a general comparison principle for interacting gases. Geom. Funct. Anal. 14, 215–244 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22, 751–756 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benamou, J.-D., Brenier, Y.: A computational Fluid Mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cordero-Erausquin, D.: Some applications of mass transport to Gaussian type inequalities. Arch. Rational Mech. Anal. 161, 257–269 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20, 253–264 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Ferone, V., Nitsch, C., Trombetti, C.: A remark on optimal weighted Poincaré inequalities for convex domains. Rend. Lincei Mat. Appl. 23, 467–475 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge, NJ (2003)

    Book  MATH  Google Scholar 

  11. Loeper, G.: Uniqueness of the solution to the Vlasov-Poisson system with bounded density. J. Math. Pures Appl. 86, 68–79 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lott, J., Villani, C.: Weak curvature conditions and functional inequalities. J. Funct. Anal. 245, 311–333 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Maggi, F., Villani, C.: Balls have the worst best Sobolev inequalities. II. Variants and extensions. Calc. Var. Partial Differ. Equ. 31, 47–74 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Maggi, F., Villani, C.: Balls have the worst best Sobolev inequalities. J. Geom. Anal. 15, 83–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Maury, B., Roudneff-Chupin, A., Santambrogio, F.: A macroscopic crowd motion model of gradient flow type. Math. Models and Methods Appl. Sci. 20, 1787–1821 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)

    Google Scholar 

  17. McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nazaret, B.: Best constant in Sobolev trace inequalities on the half space. Nonlinear Anal. 65, 1977–1985 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  20. Peyre, R.: Non-asymptotic equivalence between \(W_2\) distance and \(H^{-1}\) norm. http://arxiv.org/pdf/1104.4631v1.pdf

  21. Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differ. Equ. 44, 477–494 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Santambrogio, F.: Progress in nonlinear differential equations and their application. Optimal Transport for Applied Mathematicians, vol. 87. Birkhäuser, Basel (2015)

    Chapter  Google Scholar 

  23. Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI (2003)

    Google Scholar 

Download references

Acknowledgments

The authors wish to thank Cristina Trombetti for pointing out the references [8, 9]. This work has been partially supported by the Gaspard Monge Program for Optimization (PGMO), created by EDF and the Jacques Hadamard Mathematical Foundation, through the research contract MACRO, and by the ANR through the contract ANR-12-BS01-0014-01 GEOMETRYA.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121, Ferrara, Italy

    Lorenzo Brasco

  2. Institut de Mathématiques de Marseille, Aix-Marseille Université, 39, Rue Frédéric Joliot Curie, 13453, Marseille, France

    Lorenzo Brasco

  3. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay Cedex, France

    Filippo Santambrogio

Authors
  1. Lorenzo Brasco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filippo Santambrogio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lorenzo Brasco .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Milano, Milano, Italy

    Filippo Gazzola

  2. Mathematical Institute, Tohoku University, Sendai, Japan

    Kazuhiro Ishige

  3. Dip. di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Monte S. Angelo, Napoli, Italy

    Carlo Nitsch

  4. Dipartimento di Matematica Ulisse Dini, Università degli Studi di Firenze, Firenze, Firenze, Italy

    Paolo Salani

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Brasco, L., Santambrogio, F. (2016). A Note on Some Poincaré Inequalities on Convex Sets by Optimal Transport Methods. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_4

Download citation

Publish with us

Policies and ethics

Search

Navigation