Vector Minimizers in ℝ2

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


Let \(W:{{\mathbb R}}^m\rightarrow {{\mathbb R}}\) be a nonnegative potential with exactly two nondegenerate zeros \(a^-\neq a^+\in {{\mathbb R}}^m\). Assume that there are N ≥ 1 distinct heteroclinic orbits connecting a to a+, represented by maps \(\bar {u}_1,\ldots ,\bar {u}_N\) that minimize the one-dimensional energy \(J_{{\mathbb R}}(u)=\int _{{\mathbb R}}(\frac {\vert u^\prime \vert ^2}{2}+W(u)){d} s\). Under a nondegeneracy condition on \(\bar {u}_j\), j = 1, …, N and in two space dimensions we characterize the minimizers \(u:{{\mathbb R}}^2\rightarrow {{\mathbb R}}^m\) of the energy \({J}_\varOmega (u)=\int _\varOmega (\frac {\vert \nabla u\vert ^2}{2}+W(u)){d} x\) that converge uniformly to a± as one of the coordinates converges to ±. We prove that a bounded minimizer \(u:{{\mathbb R}}^2\rightarrow {{\mathbb R}}^m\) is necessarily an heteroclinic connection between suitable translates \(\bar {u}_-(\cdot -\eta _-)\) and \(\bar {u}_+(\cdot -\eta _+)\) of some \(\bar {u}_\pm \in \{\bar {u}_1,\ldots ,\bar {u}_N\}\). Then, assuming N = 2 and denoting \(\bar {u}_-,\bar {u}_+\) representatives of the two orbits connecting a to a+ we give a new proof of the existence (first proved in Schatzman [40]) of a solution \(u:{{\mathbb R}}^2\rightarrow {{\mathbb R}}^m\) of Δu = Wu(u), that connects certain translates of \(\bar {u}_\pm \).


  1. 1.
    Alama, S., Bronsard, L., Gui, C.: Stationary layered solutions in \({{\mathbb R}}^2\) for an Allen–Cahn system with multiple well potential. Calc. Var. 5(4), 359–390 (1997)Google Scholar
  2. 2.
    Alberti, G., Ambrosio, L., Cabré, X.: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general non linearities and a local minimality property. Acta Appl. Math. 65, 9–33 (2001)Google Scholar
  3. 3.
    Alessio, F., Montecchiari, P.: Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential. J. Fixed Point Theory Appl. 19(1), 691–717 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alikakos, N.D.: Some basic facts on the system Δu −∇W(u) = 0. Proc. Am. Math. Soc. 139, 153–162 (2011)Google Scholar
  5. 5.
    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
  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., 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
  8. 8.
    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
  9. 9.
    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
  10. 10.
    Bates, P.W., Fusco, G., Smyrnelis, P.: Entire solutions with six-fold junctions to elliptic gradient systems with triangle symmetry. Adv. Nonlinear Stud. 13(1), 1–12 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bronsard, L., Reitich, F.: On three-phase boundary motion and the singular limit of a vector-valued Ginzburg–Landau equation. Arch. Ration. Mech. Anal. 124(4), 355–379 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Busca, J., Felmer, P.: Qualitative properties of some bounded positive solutions of scalar field equations. Calc. Var. 13, 181–211 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Carr, J., Pego, B.: Metastable patterns in solutions of u t = 𝜖 2 u xx − f(u). Commun. Pure Appl. Math. 42(5), 523–576 (1989)Google Scholar
  15. 15.
    Dancer, E.: New solutions of equations in \({{\mathbb R}}^n\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 30(3–4), 535–563 (2002)Google Scholar
  16. 16.
    del Pino, M., Kowalczyk, M., Pacard, F., Wei, J.: The Toda system and multiple end solutions of autonomous planar elliptic problem. Adv. Math. 224(4), 1462–1516 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    del Pino, M., Kowalczyk, M., Pacard, F., Wei, J.: Multiple end solutions to the Allen-Cahn equation in \({{\mathbb R}}^2\). J. Differ. Geom. 258(2), 458–503 (2010)Google Scholar
  18. 18.
    del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi’s conjecture in dimension N ≥ 9. Ann. Math. 174, 1485–1569 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    del Pino, M., Kowalczyk, M., Wei, J.: Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in \({{\mathbb R}}^3\). J. Differ. Geom. 93(1), 67–131 (2013)Google Scholar
  20. 20.
    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
  21. 21.
    Farina, A.: Two results on entire solutions of Ginzburg-Landau system in higher dimensions. J. Funct. Anal. 214(2), 386–395 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Farina, A., Soave, N.: Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose–Einstein condensation. Arch. Ration. Mech. Anal. 213(1), 287–326 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Farina, A., Sciunzi, B., Valdinoci, E.: Bernstein and De Giorgi type problems: new results via a geometric approach. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5(7), 741–791 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fazly, M., Ghoussoub, N.: De Giorgi type results for elliptic systems. Calc. Var. 47(3), 809–823 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Fusco, G.: Layered solutions to the vector Allen-Cahn equation in \({{\mathbb R}}^2\). Minimizers and heteroclinic connections. Commun. Pure Appl. Math. 16(5), 1807–1841 (2017)Google Scholar
  26. 26.
    Fusco, G., Gronchi, G.F., Novaga, M.: Existence of periodic orbits near heteroclinic connections (2018).
  27. 27.
    Garofalo, N., Lin, F.H.: Unique continuation for elliptic operators: a geometric-variational approach. Commun. Pure Appl. Math. 40(3), 347–366 (1987)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Gui, C.: Hamiltonian identities for partial differential equations. J. Funct. Anal. 254(4), 904–933 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Gui, C., Schatzman, M.: Symmetric quadruple phase transitions. Indiana Univ. Math. J. 57(2), 781–836 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Gui, C., Malchiodi, A., Xu, H.: Axial symmetry of some steady state solutions to nonlinear Schrdinger equations. Proc. Am. Math. Soc. 139, 1023–1032 (2011)CrossRefGoogle Scholar
  32. 32.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)CrossRefGoogle Scholar
  33. 33.
    Liu, Y., Wang, K., Wei, J.: Global minimizers of the Allen-Cahn equation in dimension n ≥ 8. J. Math. Pures Appl. 108(6), 818–840 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Malchiodi, A.:Some new entire solutions of semilinear elliptic equations on \({{\mathbb R}}^n\). Adv. Math. 221(6), 1843–1909 (2009)Google Scholar
  35. 35.
    Monteil, A., Santambrogio, F.: Metric methods for heteroclinic connections in infinite dimensional spaces. To appear. axXiv: 1602.05487v1Google Scholar
  36. 36.
    Polácǐk, P.: Propagating terraces in a proof of the Gibbons conjecture and related results. J. Fixed Point Theory Appl. 19(1), 113–128 (2017)Google Scholar
  37. 37.
    Rudin, W.: Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York (1973)Google Scholar
  38. 38.
    Savin, O.: Regularity of flat level sets in phase transitions. Ann. Math. 169, 41–78 (2009)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Savin, O.: Minimal surfaces and minimizers of the Ginzburg landau energy. Cont. Math. Mech. Analysis AMS 526, 43–58 (2010)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Schatzman, M.: Asymmetric heteroclinic double layers. Control Optim. Calc. Var. 8 (A tribute to J. L. Lions), 965–1005 (2002, electronic)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Soave, N., Tavares, H.: New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms. J. Differ. Equ. 261(1), 505–537 (2016)CrossRefGoogle Scholar
  42. 42.
    Soave, N. Terracini, S.: Liouville theorems and 1-dimensional symmetry for solutions of an elliptic system modelling phase separation. Adv. Math. 279, 29–66 (2015)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Wei, J.: Geometrization program of semilinear elliptic equations. AMS/IP Stud. Adv. Math. 51, 831–857 (2012)MathSciNetzbMATHGoogle 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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