American Institute of Mathematical Sciences

Advanced
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
[1]

Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355
[2]

Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315
[3]

Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059
[4]

Zheng-Jian Bai, Xiao-Qing Jin, Seak-Weng Vong. On some inverse singular value problems with Toeplitz-related structure. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 187-192. doi: 10.3934/naco.2012.2.187
[5]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[6]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[7]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[8]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks & Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
[9]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks & Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009
[10]

Panos K. Palamides, Alex P. Palamides. Singular boundary value problems, via Sperner's lemma. Conference Publications, 2007, 2007 (Special) : 814-823. doi: 10.3934/proc.2007.2007.814
[11]

M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411
[12]

P. Lima, L. Morgado. Analysis of singular boundary value problems for an Emden-Fowler equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 321-336. doi: 10.3934/cpaa.2006.5.321
[13]

Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399
[14]

Stefano Biagi, Teresa Isernia. On the solvability of singular boundary value problems on the real line in the critical growth case. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1131-1157. doi: 10.3934/dcds.2020073
[15]

Luis Alvarez, Jesús Ildefonso Díaz. On the retention of the interfaces in some elliptic and parabolic nonlinear problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 1-17. doi: 10.3934/dcds.2009.25.1
[16]

Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371
[17]

Gung-Min Gie, Makram Hamouda, Roger Témam. Boundary layers in smooth curvilinear domains: Parabolic problems. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1213-1240. doi: 10.3934/dcds.2010.26.1213
[18]

Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011
[19]

Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347
[20]

Grey Ballard, John Baxley, Nisrine Libbus. Qualitative behavior and computation of multiple solutions of nonlinear boundary value problems. Communications on Pure & Applied Analysis, 2006, 5 (2) : 251-259. doi: 10.3934/cpaa.2006.5.251

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences