Communications in Mathematical Physics

, Volume 322, Issue 2, pp 515–557 | Cite as

Minimality via Second Variation for a Nonlocal Isoperimetric Problem

  • E. Acerbi
  • N. FuscoEmail author
  • M. Morini


We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L 1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.


Diblock Copolymer Volume Constraint Periodic Case Isoperimetric Problem Nonlocal Term 
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  1. 1.
    Adams, R.A., Fournier, J.F.: Sobolev Spaces (second edition) Pure and Applied Mathematics, 140. Amsterdam: Elsevier/Academic Press, 2003Google Scholar
  2. 2.
    Alberti G., Choksi R., Otto F.: Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Amer. Math. Soc. 22, 569–605 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Almgren, F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc. 4, Providence, RI: Amer. Math. Soc., 1976Google Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. New York: The Clarendon Press, Oxford University Press, 2000Google Scholar
  5. 5.
    Cagnetti F., Mora M.G., Morini M.: A second order minimality condition for the Mumford-Shah functional. Calc. Var. Part. Diff. Eqs. 33, 37–74 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Choksi R., Peletier M.A.: Small volume fraction limit of the diblock copolymer problem: I. Sharp-interface functional. SIAM J. Math. Anal. 42, 1334–1370 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Choksi R., Peletier M.A., Williams J.F.: On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional. SIAM J. Appl. Math. 69, 1712–1738 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Choksi R., Sternberg P.: Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problems. Interfaces Free Bound. 8, 371–392 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Choksi R., Sternberg P.: On the first and second variations of a nonlocal isoperimetric problem. J. Reine angew. Math. 611, 75–108 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cicalese, M., Leonardi, G.: A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206, 617–643 (2012)Google Scholar
  11. 11.
    Cicalese, M., Spadaro, E.: Droplet Minimizers of an Isoperimetric Problem with long-range interactions. Preprint, 2011, available at [math.AP], 2011
  12. 12.
    Dal Maso, G.: An Introduction to Γ-Convergence, Basel Buston: Birkhaüser, 1993Google Scholar
  13. 13.
    Esposito L., Fusco N.: A remark on a free interface problem with volume constraint. J. Convex Anal. 18, 417–426 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Figalli A., Maggi F., Pratelli A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167–211 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Fusco N., Maggi F., Pratelli A.: The sharp quantitative isoperimetric inequality. Ann. of Math. 168, 941–980 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Fusco N., Morini M.: Equilibrium configurations of epitaxially strained elastic films: second order minimality conditions and qualitative properties of solutions. Arch. Rat. Mech. Anal. 203, 247–327 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    van Gennip Y., Peletier M.A.: Stability of monolayers and bilayers in a copolymer-homopolymer blend model. Interfaces Free Bound. 11, 331–373 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Goldman, D., Muratov, C.B., Serfaty, S.: The Γ-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density. Preprint, 2012, available at [], 2012
  19. 19.
    Goldman, D., Muratov, C.B., Serfaty, S.: The Γ-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement via the renormalized energy. Preprint, 2012, available at [math.AP], 2012
  20. 20.
    Grosse-Brauckmann K.: Stable constant mean curvature surfaces minimize area. Pacific. J. Math. 175, 527–534 (1996)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hadwiger H.: Gitterperiodische Punktmengen und Isoperimetrie. Monatsh. Math. 76, 410–418 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Howards H., Hutchings M., Morgan F.: The isoperimetric problem on surfaces. Amer. Math. Monthly 106, 430–439 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Knüpfer, H., Muratov, C.B.: On an isoperimetric problem with a competing non-local term. I. The planar case. Preprint, 2011Google Scholar
  24. 24.
    Knüpfer, H., Muratov, C.B.: On an isoperimetric problem with a competing non-local term. II. The general case. Preprint, 2012.Google Scholar
  25. 25.
    Kohn R.V., Sternberg P.: Local minimisers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A 111, 69–84 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Modica L.: The gradient theory of phase transitions and minimal interface criterion. Arch. Rat. Mech. Anal. 98, 123–142 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Morgan F., Ros A.: Stable constant-mean-curvature hypersurfaces are area minimizing in small L 1 neighbourhoods. Interfaces Free Bound. 12, 151–155 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Morini, M., Sternberg, P.: Work in progress Google Scholar
  29. 29.
    Müller S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Part. Diff. Eq. 1, 169–204 (1993)zbMATHCrossRefGoogle Scholar
  30. 30.
    Muratov C.B.: Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E 66, 066108 (2002)MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Muratov C.B.: Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions. Commun. Math. Phys. 299, 45–87 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    Muratov C.B., Osipov V.V.: General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems. Phys. Rev. E 53, 3101–3116 (1996)MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Ohta T., Kawasaki K.: Equilibrium morphology of block copolymer melts. Macromolecules 19, 2621–2632 (1986)ADSCrossRefGoogle Scholar
  34. 34.
    Ren X., Wei J.: Concentrically layered energy equilibria of the di-block copolymer problem. Eur. J. Appl. Math. 13, 479–496 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Ren X., Wei J.: On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5, 193–238 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ren X., Wei J.: Stability of spot and ring solutions of the diblock copolymer equation. J. Math. Phys. 45, 4106–4133 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    Ren X., Wei J.: Wriggled lamellar solutions and their stability in the diblock copolymer problem. SIAM J. Math. Anal. 37, 455–489 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Ren X., Wei J.: Many droplet pattern in the cylindrical phase of diblock copolymer morphology. Rev. Math. Phys. 19, 879–921 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Ren X., Wei J.: Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology. SIAM J. Math. Anal. 39, 1497–1535 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Ross M.: Schwartz’ P and D surfaces are stable. Diff. Geom. Appl. 2, 179–195 (1992)zbMATHCrossRefGoogle Scholar
  41. 41.
    Schoen R., Simon L.M.: A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals. Indiana Univ. Math. J. 31, 415–434 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Simon, L.M.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, Canberra: Aust. Nat. Univ., 1983Google Scholar
  43. 43.
    Spadaro E.N.: Uniform energy and density distribution: diblock copolymers’ functional. Interfaces Free Bound. 11, 447–474 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Sternberg P., Topaloglu I.: On the global minimizers of a nonlocal isoperimetric problem in two dimensions. Interfaces Free Bound. 13, 155–169 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Tamanini I.: Boundaries of Caccioppoli sets with Hölder-continuous normal vector. J. Reine Angew. Math. 334, 27–39 (1982)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Tamanini I.: Regularity results for almost minimal oriented hypersurfaces in \({\mathbb{R}^n}\) . Quaderni del Dipartimento di Matematica dell Università di Lecce 1, 1–92 (1984)Google Scholar
  47. 47.
    Thomas E.L., Anderson D.M., Henkee C.S., Hoffman D.: Periodic area-minimizing surfaces in block copolymers. Nature 334, 598–601 (1988)ADSCrossRefGoogle Scholar
  48. 48.
    Topaloglu I.: On a nonlocal isoperimetric problem on the two-sphere. Comm. Pure Appl. Anal. 12, 597–620 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    White B.: A strong minimax property of nondegenerate minimal submanifolds. J. Reine Angew. Math. 457, 203–218 (1994)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità Degli Studi di ParmaParmaItaly
  2. 2.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università Degli Studi di Napoli “Federico II”NapoliItaly

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