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 & 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. doi: 10.1007/BF01161634. Google Scholar

[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). doi: 10.1103/PhysRevE.75.031802. Google Scholar

[3]

R. Choksi, Nonlocal Cahn-Hilliard and isoperimetric problems: Periodic phase separation induced by competing long- and short-term interactions,, in, (2008), 33. Google Scholar

[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. doi: 10.1137/090764888. Google Scholar

[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. doi: 10.1137/10079330X. Google Scholar

[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. doi: 10.1137/080728809. Google Scholar

[7]

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

[8]

M. P. do Carmo, "Differential Geometry of Curves and Surfaces,'', Prentice Hall, (1976). Google Scholar

[9]

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

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'', Birkh\, (1984). Google Scholar

[11]

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

[12]

J. Jost, "Riemannian Geometry and Geometric Analysis,'', Springer-Verlag, (2005). Google Scholar

[13]

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

[14]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbbR^n$,, Arch. Rational Mech. Anal., 55 (1974), 357. doi: 10.1007/BF00250439. Google Scholar

[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. doi: 10.1002/cpa.10113. Google Scholar

[16]

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

[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. doi: 10.1090/S0002-9947-00-02482-X. Google Scholar

[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. doi: 10.4171/IFB/230. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[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. doi: 10.1137/S0036141002413348. Google Scholar

[24]

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

[25]

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

[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. doi: 10.1137/070690286. Google Scholar

[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. doi: 10.1137/080742361. Google Scholar

[28]

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

[29]

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

[30]

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

[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. doi: 10.1017/S0308210500030110. Google Scholar

[32]

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

[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). doi: 10.1103/PhysRevE.72.016710. Google Scholar

[34]

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

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. doi: 10.1007/BF01161634. Google Scholar

[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). doi: 10.1103/PhysRevE.75.031802. Google Scholar

[3]

R. Choksi, Nonlocal Cahn-Hilliard and isoperimetric problems: Periodic phase separation induced by competing long- and short-term interactions,, in, (2008), 33. Google Scholar

[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. doi: 10.1137/090764888. Google Scholar

[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. doi: 10.1137/10079330X. Google Scholar

[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. doi: 10.1137/080728809. Google Scholar

[7]

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

[8]

M. P. do Carmo, "Differential Geometry of Curves and Surfaces,'', Prentice Hall, (1976). Google Scholar

[9]

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

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'', Birkh\, (1984). Google Scholar

[11]

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

[12]

J. Jost, "Riemannian Geometry and Geometric Analysis,'', Springer-Verlag, (2005). Google Scholar

[13]

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

[14]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbbR^n$,, Arch. Rational Mech. Anal., 55 (1974), 357. doi: 10.1007/BF00250439. Google Scholar

[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. doi: 10.1002/cpa.10113. Google Scholar

[16]

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

[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. doi: 10.1090/S0002-9947-00-02482-X. Google Scholar

[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. doi: 10.4171/IFB/230. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[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. doi: 10.1137/S0036141002413348. Google Scholar

[24]

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

[25]

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

[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. doi: 10.1137/070690286. Google Scholar

[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. doi: 10.1137/080742361. Google Scholar

[28]

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

[29]

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

[30]

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

[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. doi: 10.1017/S0308210500030110. Google Scholar

[32]

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

[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). doi: 10.1103/PhysRevE.72.016710. Google Scholar

[34]

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

[1]

Stan Alama, Lia Bronsard, Rustum Choksi, Ihsan Topaloglu. Droplet phase in a nonlocal isoperimetric problem under confinement. Communications on Pure & Applied Analysis, 2020, 19 (1) : 175-202. doi: 10.3934/cpaa.2020010

[2]

Annalisa Cesaroni, Matteo Novaga. The isoperimetric problem for nonlocal perimeters. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 425-440. doi: 10.3934/dcdss.2018023

[3]

Michael Helmers, Barbara Niethammer, Xiaofeng Ren. Evolution in off-critical diblock copolymer melts. Networks & Heterogeneous Media, 2008, 3 (3) : 615-632. doi: 10.3934/nhm.2008.3.615

[4]

Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003

[5]

Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102

[6]

Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045

[7]

Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89

[8]

Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225

[9]

Ian Johnson, Evelyn Sander, Thomas Wanner. Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671

[10]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011

[11]

Feifei Tang, Suting Wei, Jun Yang. Phase transition layers for Fife-Greenlee problem on smooth bounded domain. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1527-1552. doi: 10.3934/dcds.2018063

[12]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993

[13]

Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017

[14]

Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679

[15]

Giovanna Bonfanti, Fabio Luterotti. Global solution to a phase transition model with microscopic movements and accelerations in one space dimension. Communications on Pure & Applied Analysis, 2006, 5 (4) : 763-777. doi: 10.3934/cpaa.2006.5.763

[16]

Tatiana Odzijewicz. Generalized fractional isoperimetric problem of several variables. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2617-2629. doi: 10.3934/dcdsb.2014.19.2617

[17]

Tien-Tsan Shieh. From gradient theory of phase transition to a generalized minimal interface problem with a contact energy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2729-2755. doi: 10.3934/dcds.2016.36.2729

[18]

Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163

[19]

Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777

[20]

Alain Miranville. Some mathematical models in phase transition. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 271-306. doi: 10.3934/dcdss.2014.7.271

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]