Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 7 (2016), 1737-1772.

A double well potential system

Jaeyoung Byeon, Piero Montecchiari, and Paul H. Rabinowitz

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A semilinear elliptic system of PDEs with a nonlinear term of double well potential type is studied in a cylindrical domain. The existence of solutions heteroclinic to the bottom of the wells as minima of the associated functional is established. Further applications are given, including the existence of multitransition solutions as local minima of the functional.

Article information

Anal. PDE, Volume 9, Number 7 (2016), 1737-1772.

Received: 23 March 2016
Revised: 26 June 2016
Accepted: 30 July 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J47: Second-order elliptic systems
Secondary: 35J57: Boundary value problems for second-order elliptic systems 58E30: Variational principles

elliptic system double well potential heteroclinic minimization


Byeon, Jaeyoung; Montecchiari, Piero; Rabinowitz, Paul H. A double well potential system. Anal. PDE 9 (2016), no. 7, 1737--1772. doi:10.2140/apde.2016.9.1737.

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  • F. Alessio, “Stationary layered solutions for a system of Allen–Cahn type equations”, Indiana Univ. Math. J. 62:5 (2013), 1535–1564.
  • F. Alessio and P. Montecchiari, “Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential”, J. Differential Equations 257:12 (2014), 4572–4599.
  • N. D. Alikakos, “A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system $\Delta u-W\sb u(u)=0$”, Comm. Partial Differential Equations 37:12 (2012), 2093–2115.
  • N. D. Alikakos, “On the structure of phase transition maps for three or more coexisting phases”, pp. 1–31 in Geometric partial differential equations, CRM Series 15, Ed. Norm., Pisa, 2013.
  • N. D. Alikakos and G. Fusco, “On the connection problem for potentials with several global minima”, Indiana Univ. Math. J. 57:4 (2008), 1871–1906.
  • N. D. Alikakos and G. Fusco, “Entire solutions to nonconvex variational elliptic systems in the presence of a finite symmetry group”, pp. 1–26 in Singularities in nonlinear evolution phenomena and applications, vol. 9, CRM Series, Ed. Norm., Pisa, 2009.
  • N. D. Alikakos and G. Fusco, “Entire solutions to equivariant elliptic systems with variational structure”, Arch. Ration. Mech. Anal. 202:2 (2011), 567–597.
  • N. D. Alikakos and G. Fusco, “A maximum principle for systems with variational structure and an application to standing waves”, J. Eur. Math. Soc. $($JEMS$)$ 17:7 (2015), 1547–1567.
  • N. D. Alikakos and P. Smyrnelis, “Existence of lattice solutions to semilinear elliptic systems with periodic potential”, Electron. J. Differential Equations (2012), art. id. 15, 1–15.
  • S. V. Bolotin, “Libration motions of natural dynamical systems”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1978), 72–77.
  • L. Bronsard and F. Reitich, “On three-phase boundary motion and the singular limit of a vector-valued Ginzburg–Landau equation”, Arch. Rational Mech. Anal. 124:4 (1993), 355–379.
  • L. C. Evans, Partial differential equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin, 1983.
  • C. Gui and M. Schatzman, “Symmetric quadruple phase transitions”, Indiana Univ. Math. J. 57:2 (2008), 781–836.
  • V. V. Kozlov, “Calculus of variations in the large and classical mechanics”, Uspekhi Mat. Nauk 40:2(242) (1985), 33–60, 237. In Russian; translated in Russian Mathematical Surveys 40:2 (1985), 37–71.
  • P. Montecchiari and P. H. Rabinowitz, “On the existence of multi-transition solutions for a class of elliptic systems”, Ann. Inst. H. Poincaré Anal. Non Linéaire 33:1 (2016), 199–219.
  • P. H. Rabinowitz, “Periodic and heteroclinic orbits for a periodic Hamiltonian system”, Ann. Inst. H. Poincaré Anal. Non Linéaire 6:5 (1989), 331–346.
  • P. H. Rabinowitz, “Homoclinic and heteroclinic orbits for a class of Hamiltonian systems”, Calc. Var. Partial Differential Equations 1:1 (1993), 1–36.
  • P. H. Rabinowitz, “Solutions of heteroclinic type for some classes of semilinear elliptic partial differential equations”, J. Math. Sci. Univ. Tokyo 1:3 (1994), 525–550.
  • P. H. Rabinowitz, “Spatially heteroclinic solutions for a semilinear elliptic P.D.E.”, ESAIM Control Optim. Calc. Var. 8 (2002), 915–931.
  • P. H. Rabinowitz, “A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE”, Discrete Contin. Dyn. Syst. 10:1–2 (2004), 507–515.
  • P. H. Rabinowitz, “On a class of reversible elliptic systems”, Netw. Heterog. Media 7:4 (2012), 927–939.
  • M. Schatzman, “Asymmetric heteroclinic double layers”, ESAIM Control Optim. Calc. Var. 8 (2002), 965–1005.
  • P. Sternberg, “Vector-valued local minimizers of nonconvex variational problems”, Rocky Mountain J. Math. 21:2 (1991), 799–807.