Skip to main content
SpringerLink
Log in

Displacement Convexity and Minimal Fronts at Phase Boundaries

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables.We use this to show that, in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals and, in the multi-component setting, jointly displacement convex functionals.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alberti G.: Some remarks about a notion of rearrangement. Ann. Scuola Norm. Sup. Pisa Cl. Sci XXIX, 457–472 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Alberti G., Bellettini G.: A nonlocal anisotropic model for phase transitions Part i: the optimal profile problem. Math. Annalen. 310, 527–560 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alberti G., Bellettini G., Cassandro M., Presutti E.: Surface Tension in Ising Systems with Kac Potentials. J. Stat. Phys. 82, 743–796 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bastea S., Esposito R., Lebowitz J.L., Marra R.: Sharp interface motion of a binary fluid mixture. J. Stat. Phys. 124, 445–484 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Blower G.: Displacement convexity for the generalized orthogonal ensemble. J. Stat. Phys. 116, 1359–1387 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Brenier Y.: Polar factorization and monotone rearrangement of vector valued functions. Comm. Pure Appl. Math. 64, 375–417 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carlen E.A., Carvalho M., Esposito R., Lebowitz J.L., Marra R.: Free energy minimizers for a two-species model with segregation and liquid-vapor transition. Nonlinearity 16, 1075–1105 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Carlen E.A., Carvalho M., Esposito R., Lebowitz J.L., Marra R.: Phase transitions in equilibrium systems: microscopic models and mesoscopic free energies J. Molecular Phys. 103, 3141–3151 (2005)

    Article  ADS  Google Scholar 

  9. Dal Passo, R., De Mottoni, P.: The heat equation with a non local density dependent advection term (preprint) (1991)

  10. De Masi A., Orlandi E., Presutti E., Triolo L.: Uniqueness and global stability of the instanton in non local evolution equations. Rendiconti di Matematica 14, 693–723 (1994)

    MathSciNet  MATH  Google Scholar 

  11. De Masi A., Orlandi E., Presutti E., Triolo L.: Stability of the interface in a model of phase separation. Proc. R. Soc. Edinb. 124A, 1013–1022 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. McCann R.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. van der Waals, J.D.: Théorie moléculaire d’une substance composée de deux matières différentes, I. Verhandelingen, Kon. Akad. Wet. Amsterdam 20 (1880) and Théorie moléculaire d’une substance composée de deux matières différentes, II. Arch. Néerl. 24(1) (1891)

  14. Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Socity, Providence, 2003

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    E. A. Carlen

  2. Department of Mathematics and CMAF, University of Lisbon, 1649-003, Lisbon, Portugal

    M. C. Carvalho

  3. Dip. di Matematica, Università di L’Aquila, 67100, Coppito (AQ), Italy

    R. Esposito

  4. Departments of Mathematics and Physics, Rutgers University, New Brunswick, NJ, 08903, USA

    J. L. Lebowitz

  5. Dipartimento di Fisica and Unita’ INFN, Universita’ di Roma Tor Vergata, 00133, Rome, Italy

    R. Marra

Authors
  1. E. A. Carlen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. C. Carvalho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Esposito
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. L. Lebowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. R. Marra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Esposito.

Additional information

Communicated by Y. Brenier

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carlen, E.A., Carvalho, M.C., Esposito, R. et al. Displacement Convexity and Minimal Fronts at Phase Boundaries. Arch Rational Mech Anal 194, 823–847 (2009). https://doi.org/10.1007/s00205-008-0190-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-008-0190-9

Keywords

Search

Navigation