Abstract
The nonlocal geometric variational problem derived from the Ohta–Kawasaki diblock copolymer theory is an inhibitory system with self-organizing properties. The free energy, defined on subsets of a prescribed measure in a domain, is a sum of a local perimeter functional and a nonlocal energy given by the Green’s function of Poisson’s equation on the domain with the Neumann boundary condition. The system has the property of preventing a disc from drifting towards the domain boundary. This raises the question of whether a stationary set may have its interface touch the domain boundary. It is proved that a small, perturbed half disc exists as a stable stationary set, where the circular part of its boundary is inside the domain, as the interface, and the almost flat part of its boundary coincides with part of the domain boundary. The location of the half disc depends on two quantities: the curvature of the domain boundary, and a remnant of the Green’s function after one removes the fundamental solution and a reflection of the fundamental solution. This reflection is defined with respect to any sufficiently smooth domain boundary. It is an interesting new concept that generalizes the familiar notions of mirror image and circle inversion. When the nonlocal energy is weighted less against the local energy, the stationary half disc sits near a maximum of the curvature; when the nonlocal energy is weighted more, the half disc appears near a minimum of the remnant function. There is also an intermediate case where the half disc is near a minimum of a combination of the curvature and the remnant function.
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Acerbi E., Fusco N., Morini M.: Minimality via second variation for a nonlocal isoperimetric problem. Commun. Math. Phys. 322(2), 515–557 (2013)
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)
Bates S.F., Fredrickson G.H.: Block copolymers—designer soft materials. Phys. Today 52(2), 32–38 (1999)
Chen X., Kowalczyk M.: Existence of equilibria for the Cahn–Hilliard equation via local minimizers of the perimeter. Commun. Partial Differ. Equ. 21(7–8), 1207–1233 (1996)
Choksi R., Peletier M.A.: Small volume fraction limit of the diblock copolymer problem: I. Sharp inteface functional. SIAM J. Math. Anal. 42(3), 1334–1370 (2010)
Choksi R., Sternberg P.: On the first and second variations of a nonlocal isoperimetric problem. J. Reine Angew. Math. 611(611), 75–108 (2007)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Gilbarg D., Trudinger S.N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)
Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Boston (1984)
Goldman D., Muratov C.B., Serfaty S.: The Gamma-limit of the two-dimensional Ohta–Kawasaki energy. I. Droplet density. Arch. Ration. Mech. Anal. 210(2), 581–613 (2013)
Goldman D., Muratov C.B., Serfaty S.: The Gamma-limit of the two-dimensional Ohta–Kawasaki energy. Drop arrangement via the renormalized energy. Arch. Ration. Mech. Anal. 212(2), 445–501 (2014)
Kang X., Ren X.: Ring pattern solutions of a free boundary problem in diblock copolymer morphology. Phys. D 238(6), 645–665 (2009)
Kang X., Ren X.: The pattern of multiple rings from morphogenesis in development. J. Nonlinear Sci. 20(6), 747–779 (2010)
Morini M., Sternberg P.: Cascade of minimizers for a nonlocal isoperimetric problem in thin domains. SIAM J. Math. Anal. 46(3), 2033–2051 (2014)
Muratov C.B.: Droplet phases in non-local Ginzburg–Landau models with Coulomb repulsion in two dimensions. Commun. Math. Phys. 299(1), 45–87 (2010)
Ohta T., Kawasaki K.: Equilibrium morphology of block copolymer melts. Macromolecules 19(10), 2621–2632 (1986)
Oshita Y.: Singular limit problem for some elliptic systems. SIAM J. Math. Anal. 38(6), 1886–1911 (2007)
Ren X., Wei J.: On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31(4), 909–924 (2000)
Ren X., Wei J.: Many droplet pattern in the cylindrical phase of diblock copolymer morphology. Rev. Math. Phys. 19(8), 879–921 (2007)
Ren X., Wei J.: Single droplet pattern in the cylindrical phase of diblock copolymer morphology. J. Nonlinear Sci. 17(5), 471–503 (2007)
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)
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)
Ren X., Wei J.: Asymmetric and symmetric double bubbles in a ternary inhibitory system. SIAM J. Math. Anal. 46(4), 2798–2852 (2014)
Ren X., Wei J.: A double bubble assembly as a new phase of a ternary inhibitory system. Arch. Ration. Mech. Anal. 215(3), 967–1034 (2015)
Simon L.: Lectures on Geometric Measure Theory. Centre for Mathematical Analysis, Australian National University, Canberra (1984)
Sternberg P., Topaloglu I.: On the global minimizers of a nonlocal isoperimetric problem in two dimensions. Interfaces Free Bound. 13(1), 155–169 (2011)
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Communicated by L. Caffarelli
X. Ren and D. Shoup are supported in part by NSF grant DMS-1311856.
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Ren, X., Shoup, D. The Impact of the Domain Boundary on an Inhibitory System: Existence and Location of a Stationary Half Disc. Commun. Math. Phys. 340, 355–412 (2015). https://doi.org/10.1007/s00220-015-2451-4
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DOI: https://doi.org/10.1007/s00220-015-2451-4