# American Institute of Mathematical Sciences

April  2013, 33(4): 1407-1429. doi: 10.3934/dcds.2013.33.1407

## Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds

 1 College of Mathematics and Econometrics, Hunan University, Changsha 410082 2 Institute of Contemporary Mathematics, Henan University, School of Mathematics and Information Science, Henan University, Kaifeng 475004

Received  July 2011 Revised  August 2012 Published  October 2012

Let $(\mathcal{M}, \tilde{g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the problem $$\varepsilon^2 Δ_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0 in \mathcal{M},$$ where $\varepsilon >0$ is a small parameter and $V$ is a positive, smooth function in $\mathcal{M}$. Let $\mathcal{K}\subset \mathcal{M}$ be an $(N-1)$-dimensional smooth submanifold that divides $\mathcal{M}$ into two disjoint components $\mathcal{M}_{\pm}$. We assume $\mathcal{K}$ is stationary and non-degenerate relative to the weighted area functional $\int_{\mathcal{K}}V^{\frac{1}{2}}$. We prove that there exist two transition layer solutions $u_\varepsilon^{(1)}, u_\varepsilon^{(2)}$ when $\varepsilon$ is sufficiently small. The first layer solution $u_\varepsilon^{(1)}$ approaches $-1$ in $\mathcal{M}_{-}$ and $+1$ in $\mathcal{M}_{+}$ as $\varepsilon$ tends to 0, while the other solution $u_\varepsilon^{(2)}$ exhibits a transition layer in the opposite direction.
Citation: Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
##### References:
 [1] N. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundary,, Cal. Var. PDE, 11 (2000), 233. Google Scholar [2] S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1084. Google Scholar [3] L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces,, Math. Res. Lett., 3 (1996), 117. Google Scholar [4] E. N. Dancer and S. Yan, multi-layer solutions for an elliptic problem,, J. Diff. Eqns., 194 (2003), 382. Google Scholar [5] M. del Pino, M. Kowalczyk, F. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $R^2$,, J. Funct. Anal., 258 (2010), 458. Google Scholar [6] M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 70 (2007), 113. Google Scholar [7] M. del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interface in the Allen-Cahn equation,, Archive Rational Mechanical Analysis, 190 (2008), 141. Google Scholar [8] M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces,, Didcrete Contin. Dunam. Systems, 28 (2010), 975. doi: 10.3934/dcds.2010.28.975. Google Scholar [9] M. del Pino, M. Kowalczyk, J. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature,, Geom. Funct. Anal., 20 (2010), 918. doi: 10.1007/s00039-010-0083-6. Google Scholar [10] Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation,, J. Diff. Eqns., 238 (2007), 87. Google Scholar [11] Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation,, J. Diff. Eqns., 249 (2010), 215. Google Scholar [12] G. Flores, P. Padilla and Y. Tonegawa, Higher energy solutions in the theory of phase transitions: A variational approach,, J. Diff. Eqns., 169 (2001), 190. Google Scholar [13] R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Soc. Edinburgh, 11A (1989), 69. Google Scholar [14] M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions,, Annali di Matematica Pura et Aplicata, 184 (2005), 17. doi: 10.1007/s10231-003-0088-y. Google Scholar [15] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, Geom. Funct. Anal., 16 (2006), 924. Google Scholar [16] A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. Google Scholar [17] A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation,, Pacific J. Math., 229 (2007), 447. Google Scholar [18] A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation,, J. Fixed Point Theory Appl., 1 (2007), 305. Google Scholar [19] L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 357. Google Scholar [20] K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation,, J. Diff. Eqns., 191 (2003), 234. Google Scholar [21] K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems,, Ann. Inst. H. Poincaré Anal. NonLinéaire, 20 (2003), 107. Google Scholar [22] F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions,, J. Diff. Geom., 64 (2003), 359. Google Scholar [23] P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions,, Comm. Pure Appl. Math., 51 (1998), 551. Google Scholar [24] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I,, Commun. Pure Appl. Math., 56 (2003), 1078. Google Scholar [25] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II,, Calc. Var. Partial Differential Equations, 21 (2004), 157. Google Scholar [26] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. Google Scholar [27] J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model,, Asymptot. Anal., 69 (2010), 175. Google Scholar [28] J. Yang and X. Yang, Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation on higher dimensional domain,, Commun. Pure Appl. Anal., 1 (2013), 303. Google Scholar

show all references

##### References:
 [1] N. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundary,, Cal. Var. PDE, 11 (2000), 233. Google Scholar [2] S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,, Acta. Metall., 27 (1979), 1084. Google Scholar [3] L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces,, Math. Res. Lett., 3 (1996), 117. Google Scholar [4] E. N. Dancer and S. Yan, multi-layer solutions for an elliptic problem,, J. Diff. Eqns., 194 (2003), 382. Google Scholar [5] M. del Pino, M. Kowalczyk, F. Pacard and J. Wei, Multiple-end solutions to the Allen-Cahn equation in $R^2$,, J. Funct. Anal., 258 (2010), 458. Google Scholar [6] M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 70 (2007), 113. Google Scholar [7] M. del Pino, M. Kowalczyk and J. Wei, The Toda system and clustering interface in the Allen-Cahn equation,, Archive Rational Mechanical Analysis, 190 (2008), 141. Google Scholar [8] M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces,, Didcrete Contin. Dunam. Systems, 28 (2010), 975. doi: 10.3934/dcds.2010.28.975. Google Scholar [9] M. del Pino, M. Kowalczyk, J. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature,, Geom. Funct. Anal., 20 (2010), 918. doi: 10.1007/s00039-010-0083-6. Google Scholar [10] Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation,, J. Diff. Eqns., 238 (2007), 87. Google Scholar [11] Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation,, J. Diff. Eqns., 249 (2010), 215. Google Scholar [12] G. Flores, P. Padilla and Y. Tonegawa, Higher energy solutions in the theory of phase transitions: A variational approach,, J. Diff. Eqns., 169 (2001), 190. Google Scholar [13] R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations,, Proc. Royal Soc. Edinburgh, 11A (1989), 69. Google Scholar [14] M. Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions,, Annali di Matematica Pura et Aplicata, 184 (2005), 17. doi: 10.1007/s10231-003-0088-y. Google Scholar [15] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, Geom. Funct. Anal., 16 (2006), 924. Google Scholar [16] A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. Google Scholar [17] A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation,, Pacific J. Math., 229 (2007), 447. Google Scholar [18] A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation,, J. Fixed Point Theory Appl., 1 (2007), 305. Google Scholar [19] L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 357. Google Scholar [20] K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation,, J. Diff. Eqns., 191 (2003), 234. Google Scholar [21] K. Nakashima and K. Tanaka, Clustering layers and boundary layers in spatially inhomogeneous phase transition problems,, Ann. Inst. H. Poincaré Anal. NonLinéaire, 20 (2003), 107. Google Scholar [22] F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions,, J. Diff. Geom., 64 (2003), 359. Google Scholar [23] P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions,, Comm. Pure Appl. Math., 51 (1998), 551. Google Scholar [24] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I,, Commun. Pure Appl. Math., 56 (2003), 1078. Google Scholar [25] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II,, Calc. Var. Partial Differential Equations, 21 (2004), 157. Google Scholar [26] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. Google Scholar [27] J. Wei and J. Yang, Toda system and cluster phase transition layers in an inhomogeneous phase transition model,, Asymptot. Anal., 69 (2010), 175. Google Scholar [28] J. Yang and X. Yang, Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation on higher dimensional domain,, Commun. Pure Appl. Anal., 1 (2013), 303. Google Scholar
 [1] Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391 [2] Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303 [3] Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319 [4] Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703 [5] Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012 [6] Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823 [7] Jean-Paul Chehab, Alejandro A. Franco, Youcef Mammeri. Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 87-100. doi: 10.3934/dcdss.2017005 [8] Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 [9] Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015 [10] Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 [11] Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577 [12] Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009 [13] Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024 [14] Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077 [15] Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 [16] Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517 [17] Ken Shirakawa. Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies. Conference Publications, 2009, 2009 (Special) : 697-707. doi: 10.3934/proc.2009.2009.697 [18] Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159 [19] Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154 [20] 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

2018 Impact Factor: 1.143