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, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
References:
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References:
 [1] N. Alikakos, X. Chen and G. Fusco, Motion of a droplet by surface tension along the boundary, Cal. Var. PDE, 11 (2000), 233-306.  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), 1084-1095. Google Scholar [3] L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 117-131.  Google Scholar [4] E. N. Dancer and S. Yan, multi-layer solutions for an elliptic problem, J. Diff. Eqns., 194 (2003), 382-405.  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), 458-503.  Google Scholar [6] M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 70 (2007), 113-146.  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), 141-187.  Google Scholar [8] M. del Pino, M. Kowalczyk and J. Wei, The Jacobi-Toda system and foliated interfaces, Didcrete Contin. Dunam. Systems, A, 28 (2010), 975-1006. 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), 918-957. 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), 87-117.  Google Scholar [11] Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 249 (2010), 215-239.  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), 190-207.  Google Scholar [13] R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh, 11A (1989), 69-84.  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, (4), 184 (2005), 17-52. 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), 924-958.  Google Scholar [16] A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.  Google Scholar [17] A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation, Pacific J. Math., 229 (2007), 447-468.  Google Scholar [18] A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, J. Fixed Point Theory Appl., 1 (2007), 305-336.  Google Scholar [19] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 357-383.  Google Scholar [20] K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 191 (2003), 234-276.  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), 107-143.  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), 359-423.  Google Scholar [23] P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math., 51 (1998), 551-579.  Google Scholar [24] P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Commun. Pure Appl. Math., 56 (2003), 1078-1134.  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), 157-207.  Google Scholar [26] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.  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), 175-218.  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), 303-340. Google Scholar
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