Minimality via second variation for microphase separation of diblock copolymer melts

References

[1] E. Acerbi, N. Fusco and M. Morini, Minimality via second variation for a non-local isoperimetric problem, Comm. Math. Phys. 322 (2013), 515–557. 10.1007/s00220-013-1733-ySearch in Google Scholar

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Math. Monogr., Clarendon Press, Oxford 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar

[3] M. Bonacini and R. Cristoferi, Local and global minimality results for a nonlocal isoperimetric problem in ℝn, preprint (2013), http://arxiv.org/abs/1307.5269. Search in Google Scholar

[4] F. Cangetti, M. G. Mora and M. Morini, A second order minimality condition for the Mumford–Shah functional, Calc. Var. Partial Differential Equations 33 (2008), 37–74. 10.1007/s00526-007-0152-3Search in Google Scholar

[5] G. M. Capriani, V. Julin and G. Pisante, A quantitative second order minimality criterion for cavities in elastic bodies, SIAM J. Math. Anal. 45 (2013), no. 3, 1952–1991. 10.1137/120872152Search in Google Scholar

[6] R. Choksi and X. Ren, On a derivation of a density functional theory for microphase separation of di-block copolymers, J. Stat. Phys. 113 (2003), 151–176. 10.1023/A:1025722804873Search in Google Scholar

[7] R. Choksi and P. Sternberg, On the first and second variations of a non-local isoperimetric problem, J. reine angew. Math. 611 (2007), 75–108. 10.1515/CRELLE.2007.074Search in Google Scholar

[8] M. Cicalese and G. Leonardi, A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Anal. 206 (2012), no. 2, 617–643. 10.1007/s00205-012-0544-1Search in Google Scholar

[9] M. Cicalese and E. Spadaro, Droplet minimizers of an isoperimetric problem with long-range interactions, Comm. Pure Appl. Math. 66 (2013), no. 8, 1298–1333. 10.1002/cpa.21463Search in Google Scholar

[10] N. Fusco and M. Morini, Equilibrium configurations of epitaxially strained elastic films: Second order minimality conditions and qualitative properties of solutions, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 247–327. 10.1007/s00205-011-0451-xSearch in Google Scholar

[11] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer, New York 1983. Search in Google Scholar

[12] E. Giusti, Minimal surfaces and functions of bounded variations, Birkhäuser, Basel 1994. Search in Google Scholar

[13] M. Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Ration. Mech. Anal. 97 (1987), no. 3, 261–270. 10.1007/BF00250810Search in Google Scholar

[14] M. Grüter and J. Jost, Allard type regularity results for varifolds with free boundaries, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 13 (1986), no. 1, 129–169. Search in Google Scholar

[15] Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Phys. D 84 (1995), 31–39. 10.1016/0167-2789(95)00005-OSearch in Google Scholar

[16] T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules 19 (1986), 2621–2632. 10.1021/ma00164a028Search in Google Scholar

[17] X. Ren and J. Wei, Concentrically layered energy equilibria of the di-block copolymer problem, European J. Appl. Math. 13 (2002), 479–496. 10.1017/S0956792502004928Search in Google Scholar

[18] X. Ren and J. Wei, Stability of spot and ring solutions of the diblock copolymer equation, J. Math. Phys. 45 (2004), 4106–4133. 10.1063/1.1782280Search in Google Scholar

[19] X. Ren and J. Wei, Wriggled lamellar solutions and their stability in the diblock copolymer problem, SIAM J. Math. Anal. 37 (2005), 455–489. 10.1137/S0036141003433589Search in Google Scholar

[20] X. Ren and J. Wei, Many droplet pattern in the cylindrical phase of diblock copolymer morphology, Rev. Math. Phys. 19 (2007), 879–921. 10.1142/S0129055X07003139Search in Google Scholar

[21] X. Ren and J. Wei, Spherical solutions to a non-local free boundary problem from diblock copolymer morphology, SIAM J. Math. Anal. 39 (2008), 1497–1535. 10.1137/070690286Search in Google Scholar

[22] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. reine angew. Math. 503 (1998), 63–85. 10.1515/crll.1998.100Search in Google Scholar

[23] I. Tamanini, Boundaries of Caccioppoli sets with Höder-continuous normal vector, J. reine angew. Math. 334 (1982), 27–39. 10.1515/crll.1982.334.27Search in Google Scholar

[24] B. White, A strong minimax property of nondegenerate minimal submanifolds, J. reine angew. Math. 457 (1994), 203–218. 10.1515/crll.1994.457.203Search in Google Scholar