Skip to main content

The Ellipse Law: Kirchhoff Meets Dislocations

Abstract

In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter \({\alpha \in \mathbb{R}}\). The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for \({\alpha \in (0, 1)}\) the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes \({\sqrt{1-\alpha}}\) and \({\sqrt{1+\alpha}}\). This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrow techniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses.

References

  1. Albi G., Balagué D., Carrillo J.A., von Brecht J.: Stability analysis of flock and mill rings for second order models in swarming. SIAM J. Appl. Math. 74, 794–818 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosio L., Gigli N., Savaré G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser, Basel (2005)

    MATH  Google Scholar 

  3. Balagué D., Carrillo J.A., Laurent T., Raoul G.: Dimensionality of local minimizers of the interaction energy. Arch. Ration. Mech. Anal. 209, 1055–1088 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Balogh F., Merzi D.: Equilibrium measures for a class of potentials with discrete rotational symmetries. Constr. Approx. 42, 399–424 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Bertozzi A.L., Kolokolnikov T., Sun H., Uminsky D., von Brecht J.: Ring patterns and their bifurcations in a nonlocal model of biological swarms. Commun. Math. Sci. 13, 955–985 (2015)

    MathSciNet  MATH  Google Scholar 

  6. 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, 1140005, 39pp (2012)

    MathSciNet  MATH  Google Scholar 

  7. Blanchet, A., Carlier, G.: From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372, 20130398, 11pp (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  8. Bleher P.M., Delvaux S., Kuijlaars A.B.J.: Random matrix model with external source and a constrained vector equilibrium problem. Comm. Pure Appl. Math. 64, 116–160 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Bleher P.M., Kuijlaars A.B.J.: Orthogonal polynomials in the normal matrix model with a cubic potential. Adv. Math. 230, 1272–1321 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Brézis H., Kinderlehrer D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1974)

    MathSciNet  MATH  Google Scholar 

  11. Burbea J.: Motions of vortex patches. Lett. Math. Phys. 6, 1–16 (1982)

    ADS  MathSciNet  MATH  Google Scholar 

  12. Caffarelli L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4, 383–402 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Caffarelli L.A., Friedman A.: A singular perturbation problem for semiconductors. Boll. Un. Mat. Ital. B (7) 1, 409–421 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Caffarelli L.A., Vázquez J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Caffarelli L.A., Vázquez J.L.: Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29, 1393–1404 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Cañizo J.A., Carrillo J.A., Patacchini F.S.: Existence of compactly supported global minimisers for the interaction energy. Arch. Ration. Mech. Anal. 217, 1197–1217 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Carrillo, J.A., Castorina, D., Volzone, B.: Ground states for diffusion dominated free energies with logarithmic interaction. SIAM J. Math. Anal. 47, 1–25 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Carrillo J.A., Delgadino M.G., Mellet A.: Regularity of local minimizers of the interaction energy via obstacle problems. Commun. Math. Phys. 343, 747–781 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  19. Carrillo J.A., Figalli A., Patacchini F.S.: Geometry of minimizers for the interaction energy with mildly repulsive potentials. Ann. Inst. H. Poincaré Anal. Nonlinear 34, 1299–1308 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  20. Carrillo J.A., Huang Y.: Explicit equilibrium solutions for the aggregation equation with power-law potentials. Kinet. Relat. Models. 10, 171–192 (2017)

    MathSciNet  MATH  Google Scholar 

  21. 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. Iberoam. 19, 971–1018 (2003)

    MathSciNet  MATH  Google Scholar 

  22. Carrillo, J.A., Vázquez, J.L.: Some free boundary problems involving non-local diffusion and aggregation. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 373, 20140275, 16pp (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Dal Maso G.: An Introduction to \({\Gamma}\)-Convergence. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  24. D’Orsogna, M.R., Chuang, Y.-L., Bertozzi, A.L., Chayes, L.S.: Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302, 4pp (2006)

  25. Flierl G.R., Polvani L.M: Generalized Kirchhoff vortices. Phys. Fluids 29, 2376–2379 (1986)

    ADS  MATH  Google Scholar 

  26. Frostman O.: Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel. Lunds Univ. Mat. Sem. 3, 1–118 (1935)

    MATH  Google Scholar 

  27. Geers M.G.D., Peerlings R.H.J., Peletier M.A., Scardia L.: Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. Arch. Ration. Mech. Anal. 209, 495–539 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Hmidi T., Mateu J., Verdera J.: On rotating doubly connected vortices. J. Differ. Equ. 258, 1395–1429 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  29. Holm D.D., Putkaradze V.: Formation of clumps and patches in self-aggregation of finite-size particles. Phys. D 220, 183–196 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Kirchhoff G.: Vorlesungen über mathematische Physik. Teubner, Leipzig (1874)

    MATH  Google Scholar 

  31. Kolokolnikov, T., Sun, H., Uminsky, D., Bertozzi, A.L.: Stability of ring patterns arising from two-dimensional particle interactions. Phys. Rev. E 84, 015203, 4pp (2011)

  32. Kuijlaars A.B.J., Dragnev P.D.: Equilibrium problems associated with fast decreasing polynomials. Proc. Am. Math. Soc. 127, 1065–1074 (1999)

    MathSciNet  MATH  Google Scholar 

  33. Lamb, H.: Hydrodynamics. Cambridge University Press, Cambridge (1932)

  34. Mitchell, T.B., Rossi, L.F.: The evolution of Kirchhoff elliptic vortices. Phys. Fluids 20, 054103, 12pp (2008)

    ADS  MATH  Google Scholar 

  35. Mogilner A., Edelstein-Keshet L.: A non-local model for a swarm. J. Math. Biol. 38, 534–570 (1999)

    MathSciNet  MATH  Google Scholar 

  36. Mora M.G., Peletier M., Scardia L.: Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement. SIAM J. Math. Anal. 49, 4149–4205 (2017)

    MathSciNet  MATH  Google Scholar 

  37. Mora M.G., Rondi L., Scardia L.: The equilibrium measure for a nonlocal dislocation energy. Commun. Pure Appl. Math. 72, 136–158 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Comunm. Partial Differ. Equ. 26, 101–174 (2001)

    MathSciNet  MATH  Google Scholar 

  39. Saff E.B., Totik V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997)

    MATH  Google Scholar 

  40. Serfaty S., Vázquez J.L.: A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators. Calc. Var. Partial Differ. Equ. 49, 1091–1120 (2014)

    MathSciNet  MATH  Google Scholar 

  41. Simione R., Slepčev D., Topaloglu I.: Existence of ground states of nonlocal-interaction energies. J. Stat. Phys. 159, 972–986 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  43. Topaz C.M., Bertozzi A.L., Lewis M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68, 1601–1623 (2006)

    MathSciNet  MATH  Google Scholar 

  44. Toscani G.: One-dimensional kinetic models of granular flows. M2AN Math. Model. Numer. Anal. 34, 1277–1291 (2000)

    MathSciNet  MATH  Google Scholar 

  45. Villani C.: Topics in Optimal Transportation. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

Download references

Acknowledgements

JAC was partially supported by the Royal Society via a Wolfson Research Merit Award and by the EPSRC under the Grant EP/P031587/1. MGM and LR are partly supported by GNAMPA–INdAM. MGM acknowledges support by the European Research Council under Grant No. 290888. LR acknowledges support by the Università di Trieste through FRA 2016. LS acknowledges support by the EPSRC under the Grant EP/N035631/1. JM and JV acknowledge support by the Spanish projects MTM2013-44699 (MINECO) and MTM2016-75390 (MINECO), 2014SGR75 (Generalitat de Catalunya) and FP7-607647 (European Union).

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to J. A. Carrillo or M. G. Mora.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by C. De Lellis

Rights and permissions

OpenAccess This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carrillo, J.A., Mateu, J., Mora, M.G. et al. The Ellipse Law: Kirchhoff Meets Dislocations. Commun. Math. Phys. 373, 507–524 (2020). https://doi.org/10.1007/s00220-019-03368-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03368-w