May  2016, 15(3): 947-964. doi: 10.3934/cpaa.2016.15.947

Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian

1. 

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received  August 2015 Revised  December 2015 Published  February 2016

We study entire solutions in $R$ of the nonlocal system $(-\Delta)^{s}U+\nabla W(U)=(0,0)$ where $W:R^{2}\rightarrow R$ is a double well potential. We seek solutions $U$ which are heteroclinic in the sense that they connect at infinity a pair of global minima of $W$ and are also global minimizers. Under some symmetric assumptions on potential $W$, we prove the existence of such solutions for $s>\frac{1}{2}$, and give asymptotic behavior as $x\rightarrow\pm\infty$.
Citation: Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947
References:
[1]

S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in $\R^{2}$ for an Allen-Cahn system with multiple well potential,, \emph{Calculus of Variations and Partial Differential Equations}, 5 (1997), 359.  doi: 10.1007/s005260050071.  Google Scholar

[2]

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

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Communications in Partial Differential Equations}, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[4]

X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions,, \emph{Communications on Pure and Applied Mathematics}, 58 (2005), 1678.  doi: 10.1002/cpa.20093.  Google Scholar

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates,, \emph{Annales de l'Institut Henri Poincare (C) Non Linear Analysis}, 31 (2014), 23.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[6]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions,, \emph{Transactions of the American Mathematical Society}, 367 (2015), 911.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

[7]

X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations,, \emph{Calculus of Variations and Partial Differential Equations}, 49 (2014), 233.  doi: 10.1007/s00526-012-0580-6.  Google Scholar

[8]

X. Cabré and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian,, \emph{Discrete and Continuous Dynamical System}, 28 (2010), 1179.  doi: 10.3934/dcds.2010.28.1179.  Google Scholar

[9]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bulletin des Sciences Math\'ematiques}, 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[10]

E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations,, \emph{Communications in Partial Differential Equations}, 7 (1982), 77.  doi: 10.1080/03605308208820218.  Google Scholar

[11]

G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm,, \emph{Annali di Matematica Pura ed Applicata}, 192 (2013), 673.  doi: 10.1007/s10231-011-0243-9.  Google Scholar

[12]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Communications on Pure and Applied Mathematics}, 60 (2007), 67.  doi: 10.1002/cpa.20153.  Google Scholar

[13]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result,, \emph{Journal of Functional Analysis}, 256 (2009), 1842.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

show all references

References:
[1]

S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in $\R^{2}$ for an Allen-Cahn system with multiple well potential,, \emph{Calculus of Variations and Partial Differential Equations}, 5 (1997), 359.  doi: 10.1007/s005260050071.  Google Scholar

[2]

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

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Communications in Partial Differential Equations}, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[4]

X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions,, \emph{Communications on Pure and Applied Mathematics}, 58 (2005), 1678.  doi: 10.1002/cpa.20093.  Google Scholar

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates,, \emph{Annales de l'Institut Henri Poincare (C) Non Linear Analysis}, 31 (2014), 23.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[6]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions,, \emph{Transactions of the American Mathematical Society}, 367 (2015), 911.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

[7]

X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations,, \emph{Calculus of Variations and Partial Differential Equations}, 49 (2014), 233.  doi: 10.1007/s00526-012-0580-6.  Google Scholar

[8]

X. Cabré and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian,, \emph{Discrete and Continuous Dynamical System}, 28 (2010), 1179.  doi: 10.3934/dcds.2010.28.1179.  Google Scholar

[9]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bulletin des Sciences Math\'ematiques}, 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[10]

E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations,, \emph{Communications in Partial Differential Equations}, 7 (1982), 77.  doi: 10.1080/03605308208820218.  Google Scholar

[11]

G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm,, \emph{Annali di Matematica Pura ed Applicata}, 192 (2013), 673.  doi: 10.1007/s10231-011-0243-9.  Google Scholar

[12]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Communications on Pure and Applied Mathematics}, 60 (2007), 67.  doi: 10.1002/cpa.20153.  Google Scholar

[13]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result,, \emph{Journal of Functional Analysis}, 256 (2009), 1842.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

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