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Structure of stable solutions of a one-dimensional variational problem

Published online by Cambridge University Press:  11 October 2006

Nung Kwan Yip
Nung Kwan Yip*
Affiliation:
Department of Mathematics, Purdue University, USA; yip@math.purdue.edu
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Abstract

We prove the periodicity of all H 2-local minimizers with low energyfor a one-dimensional higher order variational problem.The results extend and complement an earlier work of Stefan Müllerwhich concerns the structure of global minimizer.The energy functional studied in this work is motivated by theinvestigation of coherent solid phase transformations and thecompetition between theeffects from regularization and formation of small scale structures.With a special choice of a bilinear double well potential function, wemake use of explicit solution formulas to analyze the intricateinteractions between the phase boundaries. Our analysis can provideinsights for tackling the problem with general potential functions.

Keywords

Higher order functionallocal minimizer.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 4 , October 2006 , pp. 721 - 751
Copyright
© EDP Sciences, SMAI, 2006

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References

Abeyaratne, R.A., Chu, C. and James, R.D., Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape. Philos. Mag. Ser. A 73 (1996) 457–497. CrossRef
Alberti, G. and Müller, S., New Approach, A to Variational Problems with Multiple Scales. Comm. Pure. Appl. Math. 54 (2001) 761–825. CrossRef
Ball, J. and James, R.D., Fine phase mixtures as minimizers of the energy. Arch. Rat. Mech. Anal. 100 (1987) 13–52. CrossRef
Ball, J., James, R.D., Proposed experimental tests of a theory of fine structures and the two-well problem. Philos. Trans. R. Soc. Lond. A 338 (1992) 389–450. CrossRef
Bates, P.W. and Xun, J., Metastable Patterns for the Cahn-Hilliard Equations, Part I. J. Diff. Eq. 111 (1994) 421–457. CrossRef
Carr, J., Gurtin, M.E. and Slemrod, M., Structured Phase Transitions on a Finite Interval. Arch. Rat. Mech. Anal. 86 (1984) 317–351. CrossRef
Carr, J. and Pego, R.L., Metastable Patterns in Solutions of $u_t = \epsilon^2 u_{xx} - f(u)$ . Comm. Pure Appl. Math. 42 (1989) 523–576. CrossRef
A.G. Khachaturyan, Theory of Structural Transformations in Solids. New York, Wiley-Interscience (1983).
Kohn, R.V. and Müller, S., Branching of twins near a austenite/twinned-martensite interface. Philos. Mag. Ser. A 66 (1992) 697–715. CrossRef
Kohn, R.V. and Müller, S., Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47 (1994) 405–435. CrossRef
Kohn, R.V. and Sternberg, P., Local minimizers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 69–84. CrossRef
Müller, S., Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. 1 (1993) 169–204. CrossRef
Ren, X., Truskinovsky, L., Finite Scale Microstructures in Nonlocal Elasticity. J. Elasticity 59 (2000) 319–355. CrossRef
Ren, X. and Wei, J., On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31 (2000) 909–924. CrossRef
Ren, X. and Wei, J., On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5 (2003) 193–238. CrossRef
Truskinovsky, L. and Zanzotto, G., Ericksen's Bar Revisited: Energy Wiggles. J. Mech. Phys. Solids 44 (1996) 1371–1408. CrossRef
Vainchtein, A., Healey, T., Rosakis, P. and Truskinovsky, L., The role of the spinodal region in one-dimensional martensitic phase transitions. Physica D 115 (1998) 29–48. CrossRef
N.K. Yip, manuscript (2005).