# Vector Minimizers in ℝ2

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

## Abstract

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 $$\bar {u}_\pm$$.

## References

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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
24. 24.
Fazly, M., Ghoussoub, N.: De Giorgi type results for elliptic systems. Calc. Var. 47(3), 809–823 (2013)
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). http://cvgmt.sns.it/paper/3882/
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)
28. 28.
Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)
29. 29.
Gui, C.: Hamiltonian identities for partial differential equations. J. Funct. Anal. 254(4), 904–933 (2008)
30. 30.
Gui, C., Schatzman, M.: Symmetric quadruple phase transitions. Indiana Univ. Math. J. 57(2), 781–836 (2008)
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)
32. 32.
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
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)
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)
39. 39.
Savin, O.: Minimal surfaces and minimizers of the Ginzburg landau energy. Cont. Math. Mech. Analysis AMS 526, 43–58 (2010)
40. 40.
Schatzman, M.: Asymmetric heteroclinic double layers. Control Optim. Calc. Var. 8 (A tribute to J. L. Lions), 965–1005 (2002, electronic)
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)
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)
43. 43.
Wei, J.: Geometrization program of semilinear elliptic equations. AMS/IP Stud. Adv. Math. 51, 831–857 (2012)

© 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