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The Pattern of Multiple Rings from Morphogenesis in Development

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Abstract

Under certain conditions the problem of morphogenesis in development and the problem of morphology in block copolymers may be reduced to one geometric problem. In two dimensions two new types of solutions are found. The first type of solution is a disconnected set of many components, each of which is close to a ring. The sizes and locations of the rings are precisely determined from the parameters and the domain shape of the problem. The solution of the second type has a coexistence pattern. Each component of the solution is either close to a ring or to a round disc. The first-type solutions are stable for certain parameter values but unstable for other values; the second-type solutions are always unstable. In both cases one establishes the equal area condition: the components in a solution all have asymptotically the same area.

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References

  • Alberti, G., Choksi, R., Otto, F.: Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Am. Math. Soc. 22(2), 569–605 (2009)

    Article  MathSciNet  Google Scholar 

  • Bates, F.S., Fredrickson, G.H.: Block copolymers—designer soft materials. Phys. Today 52(2), 32–38 (1999)

    Article  Google Scholar 

  • Chen, X., Oshita, Y.: Periodicity and uniqueness of global minimizers of an energy functional containing a long-range interaction. SIAM J. Math. Anal. 37(4), 1299–1332 (2005)

    Article  MathSciNet  Google Scholar 

  • De Giorgi, E.: Sulla convergenza di alcune successioni d’integrali del tipo dell’area. Rend. Mat. 8(6), 277–294 (1975)

    MATH  MathSciNet  Google Scholar 

  • Fife, P.C., Hilhorst, D.: The Nishiura–Ohnishi free boundary problem in the 1D case. SIAM J. Math. Anal. 33(3), 589–606 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972)

    Article  Google Scholar 

  • Kang, X., Ren, X.: Ring pattern solutions of a free boundary problem in diblock copolymer morphology. Physica D 238(6), 645–665 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Kohn, R., Sternberg, P.: Local minimisers and singular perturbations. Proc. R. Soc. Edinb. A 111(1–2), 69–84 (1989)

    MATH  MathSciNet  Google Scholar 

  • Modica, L., Mortola, S.: Un esempio di Γ −-convergenza. Boll. Union. Mat. Ital. B (5) 14(1), 285–299 (1977)

    MATH  MathSciNet  Google Scholar 

  • Nishiura, Y., Ohnishi, I.: Some mathematical aspects of the microphase separation in diblock copolymers. Physica D 84(1–2), 31–39 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Ohta, T., Kawasaki, K.: Equilibrium morphology of block copolymer melts. Macromolecules 19(10), 2621–2632 (1986)

    Article  Google Scholar 

  • Ren, X.: Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete Contin. Dyn. Syst. 24(3), 979–1003 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31(4), 909–924 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Concentrically layered energy equilibria of the di-block copolymer problem. Eur. J. Appl. Math. 13(5), 479–496 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: On energy minimizers of the di-block copolymer problem. Interfaces Free Bound. 5(2), 193–238 (2003a)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: On the spectra of 3-D lamellar solutions of the diblock copolymer problem. SIAM J. Math. Anal. 35(1), 1–32 (2003b)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Stability of spot and ring solutions of the diblock copolymer equation. J. Math. Phys. 45(11), 4106–4133 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Wriggled lamellar solutions and their stability in the diblock copolymer problem. SIAM J. Math. Anal. 37(2), 455–489 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Droplet solutions in the diblock copolymer problem with skewed monomer composition. Calc. Var. Partial Differ. Equ. 25(3), 333–359 (2006a)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Existence and stability of spherically layered solutions of the diblock copolymer equation. SIAM J. Appl. Math. 66(3), 1080–1099 (2006b)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Many droplet pattern in the cylindrical phase of diblock copolymer morphology. Rev. Math. Phys. 19(8), 879–921 (2007a)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Single droplet pattern in the cylindrical phase of diblock copolymer morphology. J. Nonlinear Sci. 17(5), 471–503 (2007b)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology. SIAM J. Math. Anal. 39(5), 1497–1535 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: Oval shaped droplet solutions in the saturation process of some pattern formation problems. SIAM J. Appl. Math. 70(4), 1120–1138 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Ren, X., Wei, J.: A toroidal tube solution to a problem of mean curvature and Newtonian potential. Preprint

  • Tolstov, G.P.: Fourier Series. Prentice-Hall, Englewood Cliffs (1962)

    Google Scholar 

  • Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237(641), 37–72 (1952)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, 430072, China

    Xiaosong Kang

  2. Department of Mathematics, The George Washington University, Washington, DC, 20052, USA

    Xiaofeng Ren

Authors
  1. Xiaosong Kang
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  2. Xiaofeng Ren
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Correspondence to Xiaofeng Ren.

Additional information

Communicated by J. Marsden.

X. Kang was supported in part by NSF of China grant 10871152.

X. Ren was supported in part by NSF grants DMS-0509725, DMS-0754066, DMS-0907777.

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Kang, X., Ren, X. The Pattern of Multiple Rings from Morphogenesis in Development. J Nonlinear Sci 20, 747–779 (2010). https://doi.org/10.1007/s00332-010-9072-z

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  • DOI: https://doi.org/10.1007/s00332-010-9072-z

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