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Variational analysis of a mesoscale model for bilayer membranes

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Abstract

We present an asymptotic analysis of a mesoscale energy for bilayer membranes that has been introduced and analyzed in two space dimensions by the second and third authors [Arch. Ration. Mech. Anal. 193 (2009), 475–537]. The energy is both nonlocal and nonconvex. It combines a surface area and a Monge–Kantorovich-distance term, leading to a competition between preferences for maximally concentrated and maximally dispersed configurations. Here we extend key results of our previous analysis to the three-dimensional case. First we prove a general lower estimate and formally identify a curvature energy in the zerothickness limit. Secondly we construct a recovery sequence and prove a matching upper-bound estimate.

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References

  1. L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

  2. Caffarelli L. A., Feldman M., McCann R. J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15, 1–26 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Feldman M., McCann R. J.: Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. Partial Differential Equations 15, 81–113 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hutchinson J. E.: Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J. 35, 45–71 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Knüpfer H., Muratov C. B.: On an isoperimetric problem with a competing nonlocal term I: The planar case. Comm. Pure Appl. Math. 66, 1129–1162 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Merlet, A highly anisotropic nonlinear elasticity model for vesicles I. Eulerian formulation, rigidity estimates and vanishing energy limit. Unpublished manuscript.

  7. B. Merlet, A highly anisotropic nonlinear elasticity model for vesicles II. Derivation of the thin bilayer bending theory. Unpublished manuscript.

  8. O. Pantz and K. Trabelsi, A new non linear shell modeling combining flexural and membrane effects. Research report, 2013.

  9. M. A. Peletier and M. Röger, Partial localization, lipid bilayers, and the elastica functional. Arch. Ration. Mech. Anal. 193 (2009), 475–537.

  10. Schätzle R.: Quadratic tilt-excess decay and strong maximum principle for varifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(3), 171–231 (2004)

    Google Scholar 

  11. Seguin B., Fried E.: Microphysical derivation of the Canham-Helfrich freeenergy density. J. Math. Biol. 68, 647–665 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica “N. Tartaglia”, Università Cattolica del Sacro Cuore, via dei Musei 41, I-25121, Brescia, Italy

    Luca Lussardi

  2. Department of Mathematics and Computer Science, Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Mark A. Peletier

  3. Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44227, Dortmund, Germany

    Matthias Röger

Authors
  1. Luca Lussardi
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  2. Mark A. Peletier
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  3. Matthias Röger
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Correspondence to Luca Lussardi.

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Dedicated to Haim Brezis

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Lussardi, L., Peletier, M.A. & Röger, M. Variational analysis of a mesoscale model for bilayer membranes. J. Fixed Point Theory Appl. 15, 217–240 (2014). https://doi.org/10.1007/s11784-014-0180-5

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  • DOI: https://doi.org/10.1007/s11784-014-0180-5

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