Communications in Mathematical Physics

, Volume 343, Issue 3, pp 747–781 | Cite as

Regularity of Local Minimizers of the Interaction Energy Via Obstacle Problems

Open Access


The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.


  1. 1.
    Albi G., Balagué D., Carrillo J.A., von Brecht J.: Stability analysis of flock and mill rings for 2nd order models in swarming. SIAM J. Appl. Math. 74, 794–818 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balagué D., Carrillo J.A., Laurent T., Raoul G.: Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability. Physica D 260, 5–25 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Balagué D., Carrillo J.A., Laurent T., Raoul G.: Dimensionality of local minimizers of the interaction energy. Arch. Rat. Mech. Anal. 209(3), 1055–1088 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Balagué D., Carrillo J.A., Yao Y.: Confinement for repulsive-attractive kernels. DCDS-B 19, 1227–1248 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bernoff A.J., Topaz C.M.: A primer of swarm equilibria. SIAM J. Appl. Dyn. Syst. 10(1), 212–250 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bertozzi A., Carrillo J.A., Laurent T.: Blowup in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22, 683–710 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bertozzi A.L., Laurent T., Léger F.: Aggregation and spreading via the newtonian potential: the dynamics of patch solutions. Math. Models Methods Appl. Sci. 22(supp01), 1140005 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Blank I.: Sharp results for the regularity and stability of the free boundary in the obstacle problem. Indiana Univ. Math. J. 50, 1077–1112 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Brézis H., Kinderlehrer D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1974)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Caffarelli L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 44, 383–402 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Caffarelli L.A.: A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets. Boll. Un. Mat. Ital. A 18.1, 109–113 (1981)MathSciNetMATHGoogle Scholar
  12. 12.
    Caffarelli L.A., Dolbeault J., Markowich P.A., Schmeiser C.: On Maxwellian equilibria of insulated semiconductors. Interfaces Free Bound. 2, 331–339 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Caffarelli L.A., Friedman A.: A singular perturbation problem for semiconductors. Boll. Un. Mat. Ital. B 7.1, 409–421 (1987)MathSciNetMATHGoogle Scholar
  14. 14.
    Caffarelli L.A., Salsa S., Silvestre L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Inventiones Mathematicae 171, 425–461 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Caffarelli L.A., Vázquez J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Caffarelli L.A., Vázquez J.L.: Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29, 1393–1404 (2011)MathSciNetMATHGoogle Scholar
  17. 17.
    Cañizo J.A., Carrillo J.A., Patacchini F.S.: Existence of global minimisers for the interaction energy. Arch. Rat. Mech. Anal. 217, 1197–1217 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Carrillo J.A., Chertock A., Huang Y.: A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. Commun. Comput. Phys. 17, 233–258 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Carrillo J.A., Chipot M., Huang Y.: On global minimizers of repulsive-attractive power-law interaction energies. Philos. Trans. R. Soc. A 372, 20130399 (2014)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Carrillo J.A., Di Francesco M., Figalli A., Laurent T., Slepčev D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156, 229–271 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Carrillo J.A., Di Francesco M., Figalli A., Laurent T., Slepčev D.: Confinement in nonlocal interaction equations. Nonlinear Anal. 75(2), 550–558 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Carrillo J.A., Ferreira L.C.F., Precioso J.C.: A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity. Adv. Math. 231(1), 306–327 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Carrillo J.A., Huang Y., Martin S.: Nonlinear stability of flock solutions in second-order swarming models. Nonlinear Anal. Real World Appl. 17, 332–343 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Carrillo J.A., McCann R.J., Villani C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19(3), 971–1018 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Carrillo J.A., McCann R.J., Villani C.: Contractions in the 2-wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179, 217–263 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Chafaï D., Gozlan N., Zitt P.-A.: First order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24, 2371–2413 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Choksi R., Fetecau R., Topaloglu I.: On minimizers of interaction functionals with competing attractive and repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 1283–1305 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    D’Orsogna, M.R., Chuang, Y., Bertozzi, A., Chayes, L.: Self-propelled particles with soft-core interactions: patterns, stability and collapse. Phys. Rev. Lett. 96, (104302) (2006)Google Scholar
  29. 29.
    Doye J.P.K., Wales D.J., Berry R.S.: The effect of the range of the potential on the structures of clusters. J. Chem. Phys. 103, 4234–4249 (1995)ADSCrossRefGoogle Scholar
  30. 30.
    Fellner K., Raoul G.: Stable stationary states of non-local interaction equations. Math. Models Methods Appl. Sci. 20(12), 2267–2291 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Fellner K., Raoul G.: Stability of stationary states of non-local equations with singular interaction potentials. Math. Comput. Model. 53(7–8), 1436–1450 (2011)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Fetecau R.C., Huang Y., Kolokolnikov T.: Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24(10), 2681–2716 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Fetecau R.C., Huang Y.: Equilibria of biological aggregations with nonlocal repulsive–attractive interactions. Physica D 260, 49–64 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Frostman, O.: Potentiel d’Equilibre et Capacité des Ensembles. Ph.D. thesis, Faculté des Sciences de Lund (1935)Google Scholar
  35. 35.
    Givens C.R., Shortt R.M.: A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31(2), 231–240 (1984)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Gustafsson B.: A simple proof of the regularity theorem for the variational inequality of the obstacle problem. Nonlinear Anal. 12(10), 1487–1490 (1986)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Hagan M.F., Chandler D.: Dynamic pathways for viral capsid assembly. Biophys. J. 91, 42–54 (2006)ADSCrossRefGoogle Scholar
  38. 38.
    Kolokolnikov T., Carrillo J.A., Bertozzi A., Fetecau R., Lewis M.: Emergent behaviour in multi-particle systems with non-local interactions. Physica D: Nonlinear Phenomena 260, 1–4 (2013)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications, vol. 88 of Pure and Applied Mathematics. Academic Press, New York-London (1980)MATHGoogle Scholar
  40. 40.
    Li H., Toscani G.: Long-time asymptotics of kinetic models of granular flows. Arch. Rat. Mech. Anal. 172(3), 407–428 (2004)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    López-García, A.: Greedy energy points with external fields. Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., vol. 507, pp. 189–207. Amer. Math. Soc., Providence, RI (2010)Google Scholar
  42. 42.
    Mattila, P.: Geometry of sets and measures in Euclidean spaces, vol. 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). Fractals and rectifiability.Google Scholar
  43. 43.
    Mogilner A., Edelstein-Keshet L.: A non-local model for a swarm. J. Math. Bio. 38, 534–570 (1999)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Mogilner A., Edelstein-Keshet L., Bent L., Spiros A.: Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol. 47(4), 353–389 (2003)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Petz, D., Hiai, F.: Logarithmic energy as an entropy functional. Advances in differential equations and mathematical physics (Atlanta, GA, 1997). Contemp. Math., vol. 217, pp. 205–221. Amer. Math. Soc., Providence, RI (1998)Google Scholar
  47. 47.
    Raoul G.: Non-local interaction equations: Stationary states and stability analysis. Differential Integral Equations 25(5-6), 417–440 (2012)MathSciNetMATHGoogle Scholar
  48. 48.
    Rechtsman, M.C., Stillinger, F.H., Torquato, S.: Optimized interactions for targeted self-assembly: application to a honeycomb lattice. Phys. Rev. Lett. 95, 22 (2005)Google Scholar
  49. 49.
    Ruelle D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin Inc, New York-Amsterdam (1969)MATHGoogle Scholar
  50. 50.
    Serfaty, S., Vázquez, J.L.: Hydrodynamic limit of nonlinear diffusions with fractional Laplacian operators. Calc. Var. PDE (2013)Google Scholar
  51. 51.
    Silvestre L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Simione R., Slepčev D., Topaloglu I.: Existence of minimizers of nonlocal interaction energies. J. Stat. Phys. 159, 972–986 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Stein E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  54. 54.
    Theil F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262, 209–236 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Villani C.: Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003)Google Scholar
  56. 56.
    von Brecht J., Uminsky D.: On soccer balls and linearized inverse statistical mechanics. J. Nonlinear Sci. 22(6), 935–959 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Wales D.J.: Energy landscapes of clusters bound by short-ranged potentials. Chem. Eur. J. Chem. Phys. 11, 2491–2494 (2010)Google Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.University of MarylandCollege ParkUSA

Personalised recommendations