American Institute of Mathematical Sciences

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March  2006, 6(2): 237-255. doi: 10.3934/dcdsb.2006.6.237

Analysis of a corner layer problem in anisotropic interfaces

N. D. Alikakos 1, , P. W. Bates 2, , J. W. Cahn 3, , P. C. Fife 4, , G. Fusco 5, and  G. B. Tanoglu 6,

1. 

Department of Mathematics, University of North Texas, Denton TX 76203, USA and University of Athens, Athens, Greece
2. 

P.W. Bates, Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States
3. 

Division of Materials Science, N.I.S.T., Gaithersburg, MD 20899, United States
4. 

Department of Mathematics, University of Utah, Salt Lake City, UT 84112, United States
5. 

Univ. degli Studi dell'Aquila, L'Aquila, Italy
6. 

Department of Mathematics, Izmir Institute of Technology, Izmir, Turkey

Received  February 2005 Revised  September 2005 Published  December 2005

We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered $Cu_3Au$ state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number $0(\frac{1}{\epsilon})$, of critical eigenvalues, where $\frac{1}{\epsilon}$ » $1$ is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues $\lambda_n = \e^{2/3}\mu_n$ with $\mu_n$ ~ $Cn^{2/3}$, as $n \to + \infty$. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.
Keywords: interfaces, singular perturbation., corner layers, stability, anisotropy, nonlinear boundary value problems, crystalline structure.
Mathematics Subject Classification: 74A50, 34B16, 34L1.
Citation: N. D. Alikakos, P. W. Bates, J. W. Cahn, P. C. Fife, G. Fusco, G. B. Tanoglu. Analysis of a corner layer problem in anisotropic interfaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 237-255. doi: 10.3934/dcdsb.2006.6.237
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