# American Institute of Mathematical Sciences

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, Calculus of Variations and Partial Differential Equations, 5 (1997), 359-390. 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, Communications on Pure and Applied Mathematics, 49 (1996), 677-715. 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, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar [4] X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Communications on Pure and Applied Mathematics, 58 (2005), 1678-1732. 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, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53. 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, Transactions of the American Mathematical Society, 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar [7] X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 233-269. 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, Discrete and Continuous Dynamical System, 28 (2010), 1179-1206. 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, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. 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, Communications in Partial Differential Equations, 7 (1982), 77-116. 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, Annali di Matematica Pura ed Applicata, 192 (2013), 673-718. 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, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. 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, Journal of Functional Analysis, 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

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##### References:
 [1] S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in $\R^{2}$ for an Allen-Cahn system with multiple well potential, Calculus of Variations and Partial Differential Equations, 5 (1997), 359-390. 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, Communications on Pure and Applied Mathematics, 49 (1996), 677-715. 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, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar [4] X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Communications on Pure and Applied Mathematics, 58 (2005), 1678-1732. 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, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53. 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, Transactions of the American Mathematical Society, 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar [7] X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 233-269. 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, Discrete and Continuous Dynamical System, 28 (2010), 1179-1206. 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, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. 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, Communications in Partial Differential Equations, 7 (1982), 77-116. 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, Annali di Matematica Pura ed Applicata, 192 (2013), 673-718. 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, Communications on Pure and Applied Mathematics, 60 (2007), 67-112. 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, Journal of Functional Analysis, 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.  Google Scholar
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