• Nicholas D. Alikakos
  • Giorgio Fusco
  • Panayotis Smyrnelis
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 91)


In this chapter we give an overview of the book. We state and motivate the main theorems and refer the reader to the appropriate sections.



N. D. A. would like to thank Alex Freire for his drawing of the tetrahedral cone in Fig. 1.2.


  1. 1.
    Alberti, G.: Variational models for phase transitions, an approach via Gamma convergence. In: Ambrosio, L., Dancer, N. (eds.) Calculus of Variations and Partial Differential Equations, pp. 95–114. Springer, Berlin (2000)CrossRefGoogle Scholar
  2. 2.
    Alikakos, N.D.: Some basic facts on the system Δu −∇W(u) = 0. Proc. Am. Math. Soc. 139, 153–162 (2011)Google Scholar
  3. 3.
    Alikakos, N.D.: A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system Δu − W u(u) = 0. Commun. Partial Differ. Equ. 37(12), 2093–2115 (2012)Google Scholar
  4. 4.
    Alikakos, N.D., Fusco, G.: On the connection problem for potentials with several global minima. Indiana Univ. Math. J. 57, 1871–1906 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Alikakos, N.D., Fusco, G.: Entire solutions to equivariant elliptic systems with variational structure. Arch. Ration. Mech. Anal. 202(2), 567–597 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alikakos, N.D., Fusco, G.: A maximum principle for systems with variational structure and an application to standing waves. J. Eur. Math. Soc. 17(7), 1547–1567 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Alikakos, N.D., Fusco, G.: Density estimates for vector minimizers and application. Discrete Contin. Dyn. Syst. 35(12), 5631–5663 (2015). Special issue edited by E. ValdinociMathSciNetCrossRefGoogle Scholar
  8. 8.
    Alikakos, N.D., Fusco, G.: Asymptotic behavior and rigidity results for symmetric solutions of the elliptic system Δu = W u(u). Annali della Scuola Normale Superiore di Pisa XV(special issue), 809–836 (2016)Google Scholar
  9. 9.
    Alikakos, N.D., Smyrnelis, P.: Existence of lattice solutions to semilinear elliptic systems with periodic potential. Electron. J. Differ. Equ. 15, 1–15 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Alikakos, N.D., Zarnescu, A.: (In preparation)Google Scholar
  11. 11.
    Alikakos, N.D., Betelú, S.I., Chen, X.: Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energies. Eur. J. Appl. Math. 17, 525–556 (2006)CrossRefGoogle Scholar
  12. 12.
    Alikakos, N.D., Katzourakis, N.: Heteroclinic travelling waves of gradient diffusion systems. Trans. Am. Math. Soc. 363, 1362–1397 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in \({\mathbb R}^3\) and a conjecture of De Giorgi. J. Am. Math. Soc. 13, 725–739 (2000)Google Scholar
  14. 14.
    André, N., Shafrir, I.: On a vector-valued singular perturbation problem on the sphere. In: Proceedings of the International Conference on Nonlinear Analysis, Recent advances in nonlinear Analysis, pp. 11–42. World Scientific Publishing, Singapore (2008)Google Scholar
  15. 15.
    Antonopoulos, P., Smyrnelis, P.: On minimizers of the Hamiltonian system u″ = ∇W(u), and on the existence of heteroclinic, homoclinic and periodic orbits. Indiana Univ. Math. J. 65(5), 1503–1524 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Baldo, S.: Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. Henri Poincaré 7(2), 67–90 (1990)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Barlow, M.T., Bass, R.F, Gui, C.: The Liouville property and a conjecture of De Giorgi. Commun. Pure Appl. Math. 53, 1007–1038 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bates, P.W., Fusco, G., Smyrnelis, P.: Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures. Arch. Ration. Mech. Anal. 225(2), 685–715 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25(1–2), 69–94 (1997)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Berestycki, H., Hamel, F., Monneau, R.: One dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J. 103(3), 375–396 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Bethuel, F., Brezis, H., Helein, F.: Ginzburg-Landau Vortices. Progress in Nonlinear Differential Equations and Their Applications, vol. 13. Birkhäuser, Basel (1994)Google Scholar
  22. 22.
    Bronsard, L., Gui, C., Schatzman, M.: A three-layered minimizer in \({\mathbb R}^2\) for a variational problem with a symmetric three-well potential. Commun. Pure. Appl. Math. 49(7), 677–715 (1996)Google Scholar
  23. 23.
    Cabré, X., Terra, J.: Saddle-shaped solutions of bistable diffusion equations in all of \({\mathbb R}^{2m}\). J. Eur. Math. Soc. 11, 819–943 (2009)Google Scholar
  24. 24.
    Caffarelli, L., Córdoba, A.: Uniform convergence of a singular perturbation problem. Commun. Pure Appl. Math. 48, 1–12 (1995)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Caffarelli, L., Córdoba, A.: Phase transitions: uniform regularity of the intermediate layers. J. Reine Angew. Math. 593, 209–235 (2006)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Caffarelli, L., Lin, F.H.: Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21(3), 847–862 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Caffarelli, L., Karakhanyan, A.L., Lin, F.H.: The geometry of solutions to a segregation problem for nondivergence systems. J. Fixed Point Theory Appl. 5, 319–351 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Colding, T.H., Minicozzi, W.P.: A Course in Minimal Surfaces. Graduate Studies in Mathematics, vol. 121. AMS, Providence (2011)Google Scholar
  29. 29.
    De Figueirdo, D.G., Magalhaes, C.A.: On nonquadratic Hamiltonian elliptic systems. Adv. Differ. Equ. 1(5), 881–898 (1996)MathSciNetzbMATHGoogle Scholar
  30. 30.
    De Giorgi, E.: Convergence problems for functionals and operators. In: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Rome (1978), pp. 131–188. Pitagora, Bologna (1979)Google Scholar
  31. 31.
    del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi’s conjecture in dimension N ≥ 9. Ann. Math. 174, 1485–1569 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Farina, A.: Symmetry for solutions of semilinear elliptic equations in \({\mathbb R}^N\) and related conjectures. Ricerche Mat. 10(Suppl 48), 129–154 (1999)Google Scholar
  33. 33.
    Farina, A.: Two results on entire solutions of Ginzburg–Landau system in higher dimensions. J. Funct. Anal. 214(2), 386–395 (2004)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Farina, A., Valdinoci, E.: The state of art for a conjecture of De Giorgi and related questions. Reaction-diffusion systems and viscosity solutions. In: Recent Progress on Rection-Diffusion Systems and Viscosity Solutions, pp. 74–96. World Scientific, Singapore (2008)Google Scholar
  35. 35.
    Fusco, G.: On some elementary properties of vector minimizers of the Allen-Cahn energy. Commun. Pure Appl. Anal. 13(3), 1045–1060 (2014)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Fusco, G.: Equivariant entire solutions to the elliptic system Δu − W u(u) = 0 for general G-invariant potentials. Calc. Var. Partial Differ. Equ. 49(3), 963–985 (2014)Google Scholar
  37. 37.
    Fusco, G.: Layered solutions to the vector Allen-Cahn equation in \({\mathbb R}^2\), minimizers and heteroclinic connections. Commun. Pure Appl. Anal. 16(5), 1807–1841 (2017)Google Scholar
  38. 38.
    Fusco, G., Gronchi, G.F., Novaga, M.: On the existence of connecting orbits for critical values of the energy. J. Differ. Equ. 263, 8848–8872 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Fusco, G., Gronchi, G.F., Novaga, M.: On the existence of heteroclinic connections. Sao Paulo J. Math. Sci. 12, 1–14 (2017)zbMATHGoogle Scholar
  40. 40.
    Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Grove, L.C., Benson, C.T.: Finite Reflection Groups. Graduate Texts in Mathematics, vol. 99, 2nd edn. Springer, Berlin (1985)CrossRefGoogle Scholar
  42. 42.
    Gui, C.: Hamiltonian identities for partial differential equations. J. Funct. Anal. 254(4), 904–933 (2008)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Gui, C., Schatzman, M.: Symmetric quadruple phase transitions. Indiana Univ. Math. J. 57(2), 781–836 (2008)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Heinze, S.: Travelling waves for semilinear parabolic partial differential equations in cylindrical domains. PhD thesis, Heidelberg University (1988)Google Scholar
  45. 45.
    Jerison, D., Monneau, R.: Towards a counter-example to a conjecture of De Giorgi in high dimensions. Ann. Mat. Pura. Appl. 183, 439–467 (2004)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Landau, L.D., Lifschitz, E.M.: Course of Theoretical Physics. Classical Field Theory, vol. 2, 4th edn. Butterworth-Heinemann, Oxford (1980)CrossRefGoogle Scholar
  47. 47.
    Lin, F., Pan, X. B., Wang, C.: Phase transition for potentials of high-dimensional wells. Commun. Pure Appl. Math. 65(6), 833–888 (2012)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Lucia, M., Muratov, C., Novaga, M.: Existence of traveling wave solutions for Ginzburg-Landau-type problems in infinite cylinders. Arch. Ration. Mech. Anal. 188(3), 475–508 (2008)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Modica, L.: The gradient theory of phase transitions and its minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)CrossRefGoogle Scholar
  50. 50.
    Modica, L.: Monotonicity of the energy for entire solutions of semilinear elliptic equations. In: Colombini, F., Marino, A., Modica, L. (eds.) Partial Differential Equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi, vol. 2, pp. 843–850. Birkhäuser, Boston (1989)Google Scholar
  51. 51.
    Modica, L., Mortola, S.: Un esempio di Γ-convergenza. Boll. Unione. Mat. Ital. Sez B 14, 285–299 (1977)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Monteil, A., Santambrogio, F.: Metric methods for heteroclinic connections. Math. Methods Appl. Sci. 41(3), 1019–1024 (2018)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69(1), 19–30 (1979)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Porter, D.A., Easterling, K.E.: Phase Transformations in Metals and Alloys, 2nd edn. Chapman and Hall, London (1996)Google Scholar
  55. 55.
    Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic hamiltonian system. Ann. Inst. Henri Poincaré 6(5), 331–346 (1989)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Risler, R.E.: Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure. Ann. Inst. Henri Poincaré Anal. Non Linear 25(2), 381–424 (2008)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Sandier, E., Serfaty, S.: Vortices in the Magnetic Ginzburg-Landau Model. Progress in Nonlinear Differential Equations and Their Applications, vol. 70. Birkhäuser, Basel (2007)Google Scholar
  58. 58.
    Savin, O.: Minimal Surfaces and Minimizers of the Ginzburg Landau energy. In: Contemporary Mathematics Mechanical Analysis AMS, vol. 526, pp. 43–58. American Mathematical Society, Providence (2010)Google Scholar
  59. 59.
    Schatzman, M.: Asymmetric heteroclinic double layers. Control Optim. Calc. Var. 8, 965–1005 (2002). A tribute to J. L. Lions (electronic)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Schoen, R.: Lecture Notes on General Relativity. Stanford University, Stanford (2009)Google Scholar
  61. 61.
    Serrin, J., Zou, H.: The existence of positive entire solutions of elliptic Hamiltonian systems. Commun. Partial Differ. Equ. 23, 577–599 (1998)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Smyrnelis, P.: Solutions to elliptic systems with mixed boundary conditions. Phd thesis (2012)Google Scholar
  63. 63.
    Souplet, P.: The proof of the Lane-Emden conjecture in four space dimensions. Adv. Math. 221, 1409–1427 (2009)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Sourdis, C.: The heteroclinic connection problem for general double-well potentials. Mediterr. J. Math. 13, 4693–4710 (2016)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Sourdis, C.: Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn Equation. Math. Methods Appl. Sci. 41(3), 966–972 (2018)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101(3), 209–260 (1988)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Sternberg, P.: Vector-valued local minimizers of nonconvex variational problems. Rocky Mountain J. Math. 21, 799–807 (1991)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Sternberg, P., Zuniga, A.: On the heteroclinic problem for multi-well gradient systems. J. Differ. Equ. 261, 3987–4007 (2016)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Taylor, J.E.: The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. Math. 103, 489–539 (1976)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Terman, D.: Infinitely many traveling wave solutions of a gradient system. Trans. Am. Math. Soc. 301(2), 537–556 (1987)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Valdinoci, E.: Plane-like minimizers in periodic media: jet flows and Ginzburg-Landau-type functionals. J. Reine Angew. Math. 574, 147–185 (2004)MathSciNetzbMATHGoogle Scholar
  72. 72.
    White, B.: Topics in geometric measure theory. Lecture notes (taken by O. Chodosh), Stanford (2012)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Nicholas D. Alikakos
    • 1
  • Giorgio Fusco
    • 2
  • Panayotis Smyrnelis
    • 3
  1. 1.Department of MathematicsNational and Kapodistrian UniversityAthensGreece
  2. 2.Department of MathematicsUniversity of L’AquilaCoppitoItaly
  3. 3.Center for Mathematical ModelingUniversity of ChileSantiagoChile

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