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Gamma Convergence of a Family of Surface–Director Bending Energies with Small Tilt

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Abstract

We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in \({\mathbb{R}^3}\) equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such types of energies arise, for example, in a model for bilayer membranes introduced by Peletier and Röger (Arch Ration Mech Anal 193(3), 475–537, 2009). Here we prove in three space dimensions in the vanishing-tilt limit a Gamma-liminf estimate with respect to a specific curvature energy. In order to obtain appropriate compactness and lower semi-continuity properties we use tools from geometric measure theory, in particular the concept of generalized Gauss graphs and curvature varifolds.

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References

  1. Allard W.K.: On the first variation of a varifold. Ann. Math. 95, 417–491 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anzellotti G., Serapioni R., Tamanini I.: Curvatures, functionals, currents. Indiana Univ. Math. J. 39(3), 617–669 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anzellotti, G., Delladio, S.: Minimization of functionals of curvatures and the Willmore problem. In: Advances in Geometric Analysis and Continuum Mechanics (Stanford, CA, 1993), Int. Press, Cambridge, pp. 33–43, 1995

  4. Canham, P.B.: The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 61, (1970)

  5. Delladio S., Scianna G.: Oriented and nonoriented curvature varifolds. Proc. R. Soc. Edinb. Sect. A 125(1), 63–83 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delladio, S.: Do generalized Gauss graphs induce curvature varifolds? Boll. Un. Mat. Ital. B (7) 10(4), 991–1017 (1996)

  7. Delladio S.: Special generalized Gauss graphs and their application to minimization of functionals involving curvatures. J. Reine Angew. Math. 486, 17–43 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York, 1969

  9. Giaquinta, M., Modica, G., Souček, J.: Cartesian currents and variational problems for mappings into spheres. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16(3), 393–485 (1990), 1989

  10. Helfrich W.: Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. Teil C 28, 693–703 (1973)

    Google Scholar 

  11. Hu, D., Zhang, P., Weinan, E.: Continuum theory of a moving membrane. Phys. Rev. E (3) 75(4), 041605, 11, (2007)

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Kuwert, E., Schätzle, R.: The Willmore functional. In: Topics in Modern Regularity Theory, volume 13 of CRM Series, pp. 1–115. Ed. Norm., Pisa (2012)

  14. Lussardi, L., Peletier, M.A., Röger, M.: Variational analysis of a mesoscale model for bilayer membranes. J. Fixed Point Theory Appl. 15(1), 217–240 (2014)

  15. Merlet, B.: A highly anisotropic nonlinear elasticity model for vesicles i. Eulerian formulation, rigidity estimates and vanishing energy limit (2013)

  16. Merlet, B.: A highly anisotropic nonlinear elasticity model for vesicles ii. Derivation of the thin bilayer bending theory, (2013)

  17. Moser R.: Towards a variational theory of phase transitions involving curvature. Proc. R. Soc. Edinb. Sect. A 142(4), 839–865 (2012)

    Article  MATH  Google Scholar 

  18. Pantz, O., Trabelsi, K.: A new non linear shell modeling combining flexural and membrane effects. (2013)

  19. Peletier M.A., Röger M.: Partial localization, lipid bilayers, and the elastica functional. Arch. Ration. Mech. Anal. 193(3), 475–537 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rivière T.: Analysis aspects of Willmore surfaces. Invent. Math. 174, 1–45 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Seguin, B., Fried, E.: Microphysical derivation of the canham-helfrich free-energy density. (2013)

  22. Serre, D.: Matrices: Theory and Applications. Graduate texts in mathematics. Springer Science+Business Media, LLC 2010

  23. Simon L.: Existence of surfaces minimizing the Willmore functional. Commun. Anal. Geom. 1, 281–326 (1993)

    MATH  Google Scholar 

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Authors and Affiliations

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

    Luca Lussardi

  2. Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44227, Dortmund, Germany

    Matthias Röger

Authors
  1. Luca Lussardi
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  2. Matthias Röger
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Correspondence to Matthias Röger.

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Communicated by S. Müller

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Lussardi, L., Röger, M. Gamma Convergence of a Family of Surface–Director Bending Energies with Small Tilt. Arch Rational Mech Anal 219, 985–1016 (2016). https://doi.org/10.1007/s00205-015-0914-6

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  • DOI: https://doi.org/10.1007/s00205-015-0914-6

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