January  2013, 12(1): 597-620. doi: 10.3934/cpaa.2013.12.597

On a nonlocal isoperimetric problem on the two-sphere

1. 

Department of Mathematics, Indiana University, Rawles Hall, 831 E 3rd St., Bloomington, IN 47405-2604, United States

Received  May 2011 Revised  September 2012 Published  September 2012

In this article we analyze the minimization of a nonlocal isoperimetric problem (NLIP) posed on the $2$-sphere. After establishing the regularity of the free boundary of minimizers, we characterize two critical points of the functional describing (NLIP): the single cap and the double cap. We show that when the parameter controlling the influence of the nonlocality is small, the single cap is not only stable but also is the global minimizer of (NLIP) for all values of the mass constraint. In other words, in this parameter regime, the global minimizer of the (NLIP) coincides with the global minimizer of the local isoperimetric problem on the 2-sphere. Furthermore, we show that in certain parameter regimes the double cap is an unstable critical point.
Citation: Ihsan Topaloglu. On a nonlocal isoperimetric problem on the two-sphere. Communications on Pure and Applied Analysis, 2013, 12 (1) : 597-620. doi: 10.3934/cpaa.2013.12.597
References:
[1]

J. L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z., 197 (1988), 123-138. doi: 10.1007/BF01161634.

[2]

T. L. Chantawansri, A. W. Bosse, A. Hexemer, H. D. Ceniceros, C. J. García-Cervera, E. J. Kramer and G. H. Fredrickson, Self-consistent field theory simulations of block copolymers assembly on a sphere, Phys. Rev. E, 75 (2007), 031802. doi: 10.1103/PhysRevE.75.031802.

[3]

R. Choksi, Nonlocal Cahn-Hilliard and isoperimetric problems: Periodic phase separation induced by competing long- and short-term interactions, in "CRM Proceedings and Lecture Notes: Singularities in PDE and the Calculus of Variations'' (eds. S. Alama, L. Bronsard and P.J. Sternberg), American Mathematical Society, (2008), 33-45.

[4]

R. Choksi and M. A. Peletier, Small volume fraction limit of the diblock copolymer problem I: Sharp interface functional, SIAM J. Math. Anal., 42 (2010), 1334-1370. doi: 10.1137/090764888.

[5]

R. Choksi and M. A. Peletier, Small volume fraction limit of the diblock copolymer problem II: Diffuse interface functional, SIAM J. Math. Anal., 43 (2011), 739-763. doi: 10.1137/10079330X.

[6]

R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM J. Appl. Math., 69 (2009), 1712-1738. doi: 10.1137/080728809.

[7]

R. Choksi and P. Sternberg, On the first and second variations of a nonlocal isoperimetric problem, J. Reine Angew. Math., 611 (2005), 75-108. doi: 10.1515/CRELLE.2007.074.

[8]

M. P. do Carmo, "Differential Geometry of Curves and Surfaces,'' Prentice Hall, New Jersey, 1976.

[9]

E. Giusti, The equilibrium configuration of liquid drops, J. Reine Angew. Math., 321 (1981), 53-63.

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Birkhäuser, Boston, 1984.

[11]

K. Grosse-Brauckmann, Stable constant mean curvature surfaces minimize area, Pacific J. Math., 175 (1996), 527-534.

[12]

J. Jost, "Riemannian Geometry and Geometric Analysis,'' Springer-Verlag, Berlin, 2005.

[13]

N. S. Landkof, "Boundations of Modern Potential Theory,'' Springer-Verlag, Berlin, 1972. doi: 10.1007/978-3-642-65183-0.

[14]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 55 (1974), 357-382. doi: 10.1007/BF00250439.

[15]

J. Montero, P. Sternberg and W. Ziemer, Local minimizers with vortices to the Ginzburg-Landau system in 3d, Comm. Pure Appl. Math., 57 (2004), 99-125. doi: 10.1002/cpa.10113.

[16]

F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc., 355 (2003), 5041-5052. doi: 10.1090/S0002-9947-03-03061-7.

[17]

F. Morgan, M. Hutchings and H. Howards, The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature, Trans. Amer. Math. Soc., 352 (2000), 4889-4909. doi: 10.1090/S0002-9947-00-02482-X.

[18]

F. Morgan and A. Ros, Stable constant-mean-curvature hypersurfaces are area minimizing in small $L^1$ neighborhoods, Interfaces Free Bound., 12 (2010), 151-155. doi: 10.4171/IFB/230.

[19]

C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), 45-87. doi: 10.1007/s00220-010-1094-8.

[20]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028.

[21]

M. A. Peletier and M. Veneroni, Stripe patterns in a model for block copolymers, Math. Model. Meth. Appl. Sci., 20 (2010), 843-907. doi: 10.1142/S0218202510004465.

[22]

X. Ren and J. Wei, On the multiplicity of two nonlocal variational problems, SIAM J. Math. Anal., 31 (2000), 909-924. doi: 10.1137/S0036141098348176.

[23]

X. Ren and J. Wei, On the spectra of three dimensional lamellar solutions of the diblock copolymer problem, SIAM J. Math. Anal., 35 (2003), 1-32. doi: 10.1137/S0036141002413348.

[24]

X. Ren and J. Wei, Wriggled lamellar solutions and their stability in the diblock copolymer problem, SIAM J. Math. Anal., 37 (2005), 455-489. doi: 10.1137/S0036141003433589.

[25]

X. Ren and J. Wei, Many droplet pattern in the cylindrical phase of diblock copolymer morphology, Rev. Math. Phys., 19 (2007), 879-921. doi: 10.1142/S0129055X07003139.

[26]

X. Ren and J. Wei, Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology, SIAM J. Math. Anal., 39 (2008), 1497-1535. doi: 10.1137/070690286.

[27]

X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Math. Anal., 70 (2009), 1120-1138. doi: 10.1137/080742361.

[28]

X. Ren and J. Wei, A toroidal tube solution of a nonlocal geometric problem, Interfaces Free Bound., 13 (2011), 127-154. doi: 10.4171/IFB/251.

[29]

M. Ritore, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, Comm. Anal. Geom., 9 (2001), 1093-1138.

[30]

P. Sternberg and I. Topaloglu, On the global minimizers of a nonlocal isoperimetric problem in two dimensions, Interfaces Free Bound., 13 (2011), 155-169. doi: 10.4171/IFB/252.

[31]

P. Sternberg and W. Ziemer, Local minimizers of a three-phase partition problem with triple junctions, Proc. Roy. Soc. Edin. Sect. A, 124 (1994), 1059-1073. doi: 10.1017/S0308210500030110.

[32]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math., 334 (1982), 27-39.

[33]

P. Tang, F. Qiu, H. Zhang and Y. Yang, Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method, Phys. Rev. E, 72 (2005), 016710. doi: 10.1103/PhysRevE.72.016710.

[34]

E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis,'' Cambridge University Press, London, 1927.

show all references

References:
[1]

J. L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z., 197 (1988), 123-138. doi: 10.1007/BF01161634.

[2]

T. L. Chantawansri, A. W. Bosse, A. Hexemer, H. D. Ceniceros, C. J. García-Cervera, E. J. Kramer and G. H. Fredrickson, Self-consistent field theory simulations of block copolymers assembly on a sphere, Phys. Rev. E, 75 (2007), 031802. doi: 10.1103/PhysRevE.75.031802.

[3]

R. Choksi, Nonlocal Cahn-Hilliard and isoperimetric problems: Periodic phase separation induced by competing long- and short-term interactions, in "CRM Proceedings and Lecture Notes: Singularities in PDE and the Calculus of Variations'' (eds. S. Alama, L. Bronsard and P.J. Sternberg), American Mathematical Society, (2008), 33-45.

[4]

R. Choksi and M. A. Peletier, Small volume fraction limit of the diblock copolymer problem I: Sharp interface functional, SIAM J. Math. Anal., 42 (2010), 1334-1370. doi: 10.1137/090764888.

[5]

R. Choksi and M. A. Peletier, Small volume fraction limit of the diblock copolymer problem II: Diffuse interface functional, SIAM J. Math. Anal., 43 (2011), 739-763. doi: 10.1137/10079330X.

[6]

R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM J. Appl. Math., 69 (2009), 1712-1738. doi: 10.1137/080728809.

[7]

R. Choksi and P. Sternberg, On the first and second variations of a nonlocal isoperimetric problem, J. Reine Angew. Math., 611 (2005), 75-108. doi: 10.1515/CRELLE.2007.074.

[8]

M. P. do Carmo, "Differential Geometry of Curves and Surfaces,'' Prentice Hall, New Jersey, 1976.

[9]

E. Giusti, The equilibrium configuration of liquid drops, J. Reine Angew. Math., 321 (1981), 53-63.

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Birkhäuser, Boston, 1984.

[11]

K. Grosse-Brauckmann, Stable constant mean curvature surfaces minimize area, Pacific J. Math., 175 (1996), 527-534.

[12]

J. Jost, "Riemannian Geometry and Geometric Analysis,'' Springer-Verlag, Berlin, 2005.

[13]

N. S. Landkof, "Boundations of Modern Potential Theory,'' Springer-Verlag, Berlin, 1972. doi: 10.1007/978-3-642-65183-0.

[14]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 55 (1974), 357-382. doi: 10.1007/BF00250439.

[15]

J. Montero, P. Sternberg and W. Ziemer, Local minimizers with vortices to the Ginzburg-Landau system in 3d, Comm. Pure Appl. Math., 57 (2004), 99-125. doi: 10.1002/cpa.10113.

[16]

F. Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc., 355 (2003), 5041-5052. doi: 10.1090/S0002-9947-03-03061-7.

[17]

F. Morgan, M. Hutchings and H. Howards, The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature, Trans. Amer. Math. Soc., 352 (2000), 4889-4909. doi: 10.1090/S0002-9947-00-02482-X.

[18]

F. Morgan and A. Ros, Stable constant-mean-curvature hypersurfaces are area minimizing in small $L^1$ neighborhoods, Interfaces Free Bound., 12 (2010), 151-155. doi: 10.4171/IFB/230.

[19]

C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), 45-87. doi: 10.1007/s00220-010-1094-8.

[20]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028.

[21]

M. A. Peletier and M. Veneroni, Stripe patterns in a model for block copolymers, Math. Model. Meth. Appl. Sci., 20 (2010), 843-907. doi: 10.1142/S0218202510004465.

[22]

X. Ren and J. Wei, On the multiplicity of two nonlocal variational problems, SIAM J. Math. Anal., 31 (2000), 909-924. doi: 10.1137/S0036141098348176.

[23]

X. Ren and J. Wei, On the spectra of three dimensional lamellar solutions of the diblock copolymer problem, SIAM J. Math. Anal., 35 (2003), 1-32. doi: 10.1137/S0036141002413348.

[24]

X. Ren and J. Wei, Wriggled lamellar solutions and their stability in the diblock copolymer problem, SIAM J. Math. Anal., 37 (2005), 455-489. doi: 10.1137/S0036141003433589.

[25]

X. Ren and J. Wei, Many droplet pattern in the cylindrical phase of diblock copolymer morphology, Rev. Math. Phys., 19 (2007), 879-921. doi: 10.1142/S0129055X07003139.

[26]

X. Ren and J. Wei, Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology, SIAM J. Math. Anal., 39 (2008), 1497-1535. doi: 10.1137/070690286.

[27]

X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Math. Anal., 70 (2009), 1120-1138. doi: 10.1137/080742361.

[28]

X. Ren and J. Wei, A toroidal tube solution of a nonlocal geometric problem, Interfaces Free Bound., 13 (2011), 127-154. doi: 10.4171/IFB/251.

[29]

M. Ritore, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, Comm. Anal. Geom., 9 (2001), 1093-1138.

[30]

P. Sternberg and I. Topaloglu, On the global minimizers of a nonlocal isoperimetric problem in two dimensions, Interfaces Free Bound., 13 (2011), 155-169. doi: 10.4171/IFB/252.

[31]

P. Sternberg and W. Ziemer, Local minimizers of a three-phase partition problem with triple junctions, Proc. Roy. Soc. Edin. Sect. A, 124 (1994), 1059-1073. doi: 10.1017/S0308210500030110.

[32]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math., 334 (1982), 27-39.

[33]

P. Tang, F. Qiu, H. Zhang and Y. Yang, Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method, Phys. Rev. E, 72 (2005), 016710. doi: 10.1103/PhysRevE.72.016710.

[34]

E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis,'' Cambridge University Press, London, 1927.

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