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 and Continuous Dynamical Systems, 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), 233-306.

[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.

[3]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 117-131.

[4]

E. N. Dancer and S. Yan, multi-layer solutions for an elliptic problem, J. Diff. Eqns., 194 (2003), 382-405.

[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.

[6]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 70 (2007), 113-146.

[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.

[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.

[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.

[10]

Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation, J. Diff. Eqns., 238 (2007), 87-117.

[11]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 249 (2010), 215-239.

[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.

[13]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh, 11A (1989), 69-84.

[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.

[15]

F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16 (2006), 924-958.

[16]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.

[17]

A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation, Pacific J. Math., 229 (2007), 447-468.

[18]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, J. Fixed Point Theory Appl., 1 (2007), 305-336.

[19]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 357-383.

[20]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 191 (2003), 234-276.

[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.

[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.

[23]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math., 51 (1998), 551-579.

[24]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Commun. Pure Appl. Math., 56 (2003), 1078-1134.

[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.

[26]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.

[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.

[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.

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), 233-306.

[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.

[3]

L. Bronsard and B. Stoth, On the existence of high multiplicity interfaces, Math. Res. Lett., 3 (1996), 117-131.

[4]

E. N. Dancer and S. Yan, multi-layer solutions for an elliptic problem, J. Diff. Eqns., 194 (2003), 382-405.

[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.

[6]

M. del Pino, M. Kowalczyk and J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 70 (2007), 113-146.

[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.

[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.

[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.

[10]

Y. Du and K. Nakashima, Morse index of layered solutions to the heterogeneous Allen-Cahn equation, J. Diff. Eqns., 238 (2007), 87-117.

[11]

Z. Du and C. Gui, Interior layers for an inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 249 (2010), 215-239.

[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.

[13]

R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh, 11A (1989), 69-84.

[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.

[15]

F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16 (2006), 924-958.

[16]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.

[17]

A. Malchiodi, W.-M. Ni and J. Wei, Boundary clustered interfaces for the Allen-Cahn equation, Pacific J. Math., 229 (2007), 447-468.

[18]

A. Malchiodi and J. Wei, Boundary interface for the Allen-Cahn equation, J. Fixed Point Theory Appl., 1 (2007), 305-336.

[19]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 357-383.

[20]

K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Diff. Eqns., 191 (2003), 234-276.

[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.

[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.

[23]

P. Padilla and Y. Tonegawa, On the convergence of stable phase transitions, Comm. Pure Appl. Math., 51 (1998), 551-579.

[24]

P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, I, Commun. Pure Appl. Math., 56 (2003), 1078-1134.

[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.

[26]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.

[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.

[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.

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