September  2017, 16(5): 1807-1841. doi: 10.3934/cpaa.2017088

Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections

University of L'Aquila, DISIM, via Vetoio, Coppito, 67010 L'Aquila, Italy

 

Received  November 2016 Revised  March 2017 Published  May 2017

Let
$W:\mathbb{R}^m\to \mathbb{R}$
be a nonnegative potential with exactly two nondegenerate zeros
$a_-≠ a_+∈\mathbb{R}^m$
. We assume that there are
$N≥q 1$
distinct heteroclinic orbits connecting
$a_-$
to
$a_+$
represented by maps
$\bar{u}_1,...,\bar{u}_N$
that minimize the one-dimensional energy
$J_\mathbb{R}(u)=∈t_\mathbb{R}(\frac{\vert u^\prime\vert^2}{2}+W(u)){d} s$
.
We first consider the problem of characterizing the minimizers
$u:\mathbb{R}^n\to \mathbb{R}^m$
of the energy
$\mathcal{J}_Ω(u)=∈t_Ω(\frac{\vert\nabla u\vert^2}{2}+W(u)){d} x$
. Under a nondegeneracy condition on
$\bar{u}_j$
,
$j=1,...,N$
and in two space dimensions, we prove that, provided it remains away from
$a_-$
and
$a_+$
in corresponding half spaces
$S_-$
and
$S_+$
, a bounded minimizer
$u:\mathbb{R}^2\to \mathbb{R}^m$
is necessarily an heteroclinic connection between suitable translates
$\bar{u}_-(·-η_-)$
and
$\bar{u}_+(·-η_+)$
of some
$\bar{u}_±∈\{\bar{u}_1,...,\bar{u}_N\}$
.
Then we focus on the existence problem and assuming
$N=2$
and denoting
$\bar{u}_-,\bar{u}_+$
the representations of the two orbits connecting
$a_-$
to
$a_+$
we give a new proof of the existence (first proved in [32]) of a solution
$u:\mathbb{R}^2\to \mathbb{R}^m$
of
$Δ u=W_u(u),$
that connects certain translates of
$\bar{u}_±$
.
Citation: Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088
References:
[1]

S. AlamaL. Bronsard and C. Gui, Stationary layered solutions in $\mathbb{R}^2$ for an Allen-Cahn system with multiple well potential, Calc. Var., 5 (1997), 359-390.  doi: 10.1007/s005260050071.  Google Scholar

[2]

S. Alama and Y. Li, On ''multibamp" bound states for certain semilinear elliptic equations, Ind. Uni. Math. Jour., 41 (1992), 983-1026.  doi: 10.1512/iumj.1992.41.41052.  Google Scholar

[3]

G. AlbertiL. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: simmetry in 3D for general non linearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33.  doi: 10.1023/A:1010602715526.  Google Scholar

[4]

N. D. Alikakos, Some basic facts on the system $Δ u-W_u(u)=0$, Proc. Amer. Math. Soc., 139 (2011), 153-162.  doi: 10.1090/S0002-9939-2010-10453-7.  Google Scholar

[5]

N. D. AlikakosS. I. Betelú and X. Chen, Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energies, Eur. J. Appl. Math., 17 (2006), 525-556.  doi: 10.1017/S095679250600667X.  Google Scholar

[6]

N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906.  doi: 10.1512/iumj.2008.57.3181.  Google Scholar

[7]

N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves, Journal of the European Mathematical Society, 17 (2015), 1547-1567.  doi: 10.4171/JEMS/538.  Google Scholar

[8]

N. D. Alikakos and G. Fusco, Density estimates for vector minimizers and applications, Discr. Cont. Dynam. Syst., 35 (2015), 5631-5663.  doi: 10.3934/dcds.2015.35.5631.  Google Scholar

[9]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi, Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[10]

P. W. Bates and X. Ren, Transition layers solutions of a higher order equation in an infinite tube, Comm. Part. Diff. Equat., 21 (1996), 195-220.  doi: 10.1080/03605309608821180.  Google Scholar

[11]

P. W. Bates, G. Fusco and P. Smyrnelis, Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures, preprint, arXiv: 1411.4008. Google Scholar

[12]

L. BronsardC. Gui and M. Schatzman, A three-layered minimizer in $\mathbb{R}^2$ for a variational problem with a symmetric three-well potential, Comm. Pure. Appl. Math., 49 (1996), 677-715.  doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.3.CO;2-6.  Google Scholar

[13]

L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rat. Mech. Anal., 124 (1993), 355-379.  doi: 10.1007/BF00375607.  Google Scholar

[14]

J. Carr and B. Pego, Metastable patterns in solutions of $u_t=ε^2 u_{xx}-f(u)$, Comm. Pure. Appl. Math., 42 (1989), 523-576.  doi: 10.1002/cpa.3160420502.  Google Scholar

[15]

M. Del PinoM. Kowalczyk and J. Wei, On De Giorgi's conjecture in dimension $N≥q 9$, Annal. Math., 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3.  Google Scholar

[16]

A. Farina, Symmetry for solutions of semilinear elliptic equations in $\mathbb{R}^N$ and related conjectures, Rend. Mat. Acc. Lincei., 10 (1999), 255-265.   Google Scholar

[17]

A. Farina, On the classification of entire local minimizers of the Ginzburg-Landau equation, Contemp. Math., 595 (2013), 231-236.  doi: 10.1090/conm/595/11786.  Google Scholar

[18]

A. FarinaB. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 741-791.   Google Scholar

[19]

M. Fazly and N. Ghoussouby, De Giorgi type results for elliptic systems, Calc. Var. PDE, 47 (2013), 809-823.  doi: 10.1007/s00526-012-0536-x.  Google Scholar

[20]

G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060.  doi: 10.3934/cpaa.2014.13.1045.  Google Scholar

[21]

N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights an unique continuation, Indiana Univ. Math. J., 35 (1986), 245-267.  doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[22]

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.  doi: 10.1007/s002080050196.  Google Scholar

[23]

N. Ghoussoub and C. Gui, On De Giorgi's conjecture in dimensions 4 and 5, Ann. Math., 157 (2003), 313-334.  doi: 10.4007/annals.2003.157.313.  Google Scholar

[24]

C. Gui, Hamiltonian identities for elliptic differential equations, J. Funct. Anal., 254 (2008), 904-933.  doi: 10.1016/j.jfa.2007.10.015.  Google Scholar

[25]

C. GuiA. Malchiodi and H. Xu, Axial symmetry of some steady state solutions to nonlinear Schrdinger equations, Proc. Amer. Math. Soc., 139 (2011), 1023-1032.  doi: 10.1090/S0002-9939-2010-10638-X.  Google Scholar

[26]

C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Ind. Univ. Math. J., 57 (2008), 781-836.  doi: 10.1512/iumj.2008.57.3089.  Google Scholar

[27]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. 840,1980.  Google Scholar

[28]

A. Monteil and F. Santambrogio, Metric methods for heteroclinic connections Mathematical Methods in the Applied Sciences. doi: 10.1002/mma.4072.  Google Scholar

[29]

P. Rabinowitz, Solutions of heteroclinic type for some classes of semilinear elliptic partial differential equations, Jour. Math. Sci. Uni. Tokyo, 2 (1994), 525-550.   Google Scholar

[30]

W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Math. , 1973.  Google Scholar

[31]

O. Savin, Regularity of level sets in phase transitions, Ann. Math., 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41.  Google Scholar

[32]

M. Schatzman, Asymmetric heteroclinic double layers, ESAIM Control Optim. Calc. Var. , 8 (2002), 965–1005 (A tribute to J. L. Lions). doi: 10.1051/cocv:2002039.  Google Scholar

[33]

N. Soave and S. Terracini, Liouville theorems and 1-dimensional symmetry for solutions of an elliptic system modeling phase separation, Adv. Math., 279 (2015), 29-66.  doi: 10.1016/j.aim.2015.03.015.  Google Scholar

[34]

C. Sourdis, The heteroclinic connection problem for general double-well potentials, Mediterranean Journal of Mathematics, 13 (2016), 4693-4710.  doi: 10.1007/s00009-016-0770-0.  Google Scholar

[35]

A. Zuniga and P. Sternberg, On the heteroclinic connection problem for multi-well gradient systems, Journal of Differential Equations, 261 (2016), 3987-4007.  doi: 10.1016/j.jde.2016.06.010.  Google Scholar

show all references

References:
[1]

S. AlamaL. Bronsard and C. Gui, Stationary layered solutions in $\mathbb{R}^2$ for an Allen-Cahn system with multiple well potential, Calc. Var., 5 (1997), 359-390.  doi: 10.1007/s005260050071.  Google Scholar

[2]

S. Alama and Y. Li, On ''multibamp" bound states for certain semilinear elliptic equations, Ind. Uni. Math. Jour., 41 (1992), 983-1026.  doi: 10.1512/iumj.1992.41.41052.  Google Scholar

[3]

G. AlbertiL. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: simmetry in 3D for general non linearities and a local minimality property, Acta Appl. Math., 65 (2001), 9-33.  doi: 10.1023/A:1010602715526.  Google Scholar

[4]

N. D. Alikakos, Some basic facts on the system $Δ u-W_u(u)=0$, Proc. Amer. Math. Soc., 139 (2011), 153-162.  doi: 10.1090/S0002-9939-2010-10453-7.  Google Scholar

[5]

N. D. AlikakosS. I. Betelú and X. Chen, Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energies, Eur. J. Appl. Math., 17 (2006), 525-556.  doi: 10.1017/S095679250600667X.  Google Scholar

[6]

N. D. Alikakos and G. Fusco, On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906.  doi: 10.1512/iumj.2008.57.3181.  Google Scholar

[7]

N. D. Alikakos and G. Fusco, A maximum principle for systems with variational structure and an application to standing waves, Journal of the European Mathematical Society, 17 (2015), 1547-1567.  doi: 10.4171/JEMS/538.  Google Scholar

[8]

N. D. Alikakos and G. Fusco, Density estimates for vector minimizers and applications, Discr. Cont. Dynam. Syst., 35 (2015), 5631-5663.  doi: 10.3934/dcds.2015.35.5631.  Google Scholar

[9]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $R^3$ and a conjecture of De Giorgi, Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[10]

P. W. Bates and X. Ren, Transition layers solutions of a higher order equation in an infinite tube, Comm. Part. Diff. Equat., 21 (1996), 195-220.  doi: 10.1080/03605309608821180.  Google Scholar

[11]

P. W. Bates, G. Fusco and P. Smyrnelis, Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures, preprint, arXiv: 1411.4008. Google Scholar

[12]

L. BronsardC. Gui and M. Schatzman, A three-layered minimizer in $\mathbb{R}^2$ for a variational problem with a symmetric three-well potential, Comm. Pure. Appl. Math., 49 (1996), 677-715.  doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.3.CO;2-6.  Google Scholar

[13]

L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rat. Mech. Anal., 124 (1993), 355-379.  doi: 10.1007/BF00375607.  Google Scholar

[14]

J. Carr and B. Pego, Metastable patterns in solutions of $u_t=ε^2 u_{xx}-f(u)$, Comm. Pure. Appl. Math., 42 (1989), 523-576.  doi: 10.1002/cpa.3160420502.  Google Scholar

[15]

M. Del PinoM. Kowalczyk and J. Wei, On De Giorgi's conjecture in dimension $N≥q 9$, Annal. Math., 174 (2011), 1485-1569.  doi: 10.4007/annals.2011.174.3.3.  Google Scholar

[16]

A. Farina, Symmetry for solutions of semilinear elliptic equations in $\mathbb{R}^N$ and related conjectures, Rend. Mat. Acc. Lincei., 10 (1999), 255-265.   Google Scholar

[17]

A. Farina, On the classification of entire local minimizers of the Ginzburg-Landau equation, Contemp. Math., 595 (2013), 231-236.  doi: 10.1090/conm/595/11786.  Google Scholar

[18]

A. FarinaB. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 741-791.   Google Scholar

[19]

M. Fazly and N. Ghoussouby, De Giorgi type results for elliptic systems, Calc. Var. PDE, 47 (2013), 809-823.  doi: 10.1007/s00526-012-0536-x.  Google Scholar

[20]

G. Fusco, On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060.  doi: 10.3934/cpaa.2014.13.1045.  Google Scholar

[21]

N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights an unique continuation, Indiana Univ. Math. J., 35 (1986), 245-267.  doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[22]

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491.  doi: 10.1007/s002080050196.  Google Scholar

[23]

N. Ghoussoub and C. Gui, On De Giorgi's conjecture in dimensions 4 and 5, Ann. Math., 157 (2003), 313-334.  doi: 10.4007/annals.2003.157.313.  Google Scholar

[24]

C. Gui, Hamiltonian identities for elliptic differential equations, J. Funct. Anal., 254 (2008), 904-933.  doi: 10.1016/j.jfa.2007.10.015.  Google Scholar

[25]

C. GuiA. Malchiodi and H. Xu, Axial symmetry of some steady state solutions to nonlinear Schrdinger equations, Proc. Amer. Math. Soc., 139 (2011), 1023-1032.  doi: 10.1090/S0002-9939-2010-10638-X.  Google Scholar

[26]

C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Ind. Univ. Math. J., 57 (2008), 781-836.  doi: 10.1512/iumj.2008.57.3089.  Google Scholar

[27]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. 840,1980.  Google Scholar

[28]

A. Monteil and F. Santambrogio, Metric methods for heteroclinic connections Mathematical Methods in the Applied Sciences. doi: 10.1002/mma.4072.  Google Scholar

[29]

P. Rabinowitz, Solutions of heteroclinic type for some classes of semilinear elliptic partial differential equations, Jour. Math. Sci. Uni. Tokyo, 2 (1994), 525-550.   Google Scholar

[30]

W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Math. , 1973.  Google Scholar

[31]

O. Savin, Regularity of level sets in phase transitions, Ann. Math., 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41.  Google Scholar

[32]

M. Schatzman, Asymmetric heteroclinic double layers, ESAIM Control Optim. Calc. Var. , 8 (2002), 965–1005 (A tribute to J. L. Lions). doi: 10.1051/cocv:2002039.  Google Scholar

[33]

N. Soave and S. Terracini, Liouville theorems and 1-dimensional symmetry for solutions of an elliptic system modeling phase separation, Adv. Math., 279 (2015), 29-66.  doi: 10.1016/j.aim.2015.03.015.  Google Scholar

[34]

C. Sourdis, The heteroclinic connection problem for general double-well potentials, Mediterranean Journal of Mathematics, 13 (2016), 4693-4710.  doi: 10.1007/s00009-016-0770-0.  Google Scholar

[35]

A. Zuniga and P. Sternberg, On the heteroclinic connection problem for multi-well gradient systems, Journal of Differential Equations, 261 (2016), 3987-4007.  doi: 10.1016/j.jde.2016.06.010.  Google Scholar

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