Singular perturbations as a selection criterion for periodic minimizing sequences

  • Stefan Müller
Article

Summary

Minimizers of functionals like\(\int_0^1 { \in ^2 u^2 _{xx} } + (u_x^2 - 1)^2 + u^2 dx\) subject to periodic (or Dirichlet) boundary conditions are investigated. While for ε=0 the infimum is not attained it is shown that for sufficiently small ε > 0, all minimizers are periodic with period ∼ ε1/3. Connections with solid-solid phase transformations are indicated.

Mathematics subject classification

34E15 49J45 (34C25, 35M10) 

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References

  1. [Ag 82]
    Agmon, S.: Exponential decay of solutions of second order elliptic equations, Princeton: Princeton University Press 1982Google Scholar
  2. [ATL 85]
    Amelincks, S., Tendeloo, G. van, Landuyt, J. van: EM and ED studies of phase transformations and phase stability. Mater. Sci. Forum3, 123–145 (1985)Google Scholar
  3. [Ar 90]
    Arlt, G.: Twinning in ferroelectric and ferroelastic ceramics: stress relief. J. Mater. Sci.22, 2655–2666 (1990)Google Scholar
  4. [At 84]
    Attouch, H.: Variational convergence of functions and operators. London: Pitman 1984Google Scholar
  5. [AG 87]
    Aviles, P., Giga, A.: A mathematical problem related to the physical theory of liquid crystals. Proc. Cent. Math. Anal. Aust. Natl. Univ.12, 1–16 (1987)Google Scholar
  6. [AG 89]
    Singularities and rank-1 properties of Hessian measures. Duke Math. J.58, 441–467 (1989)Google Scholar
  7. [Ba 91]
    Baldo, S.: Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. Henri Poincaré, Anal. Nonlinéaire (to appear)Google Scholar
  8. [BJ 87]
    Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of the energy. Arch. Rat. Mech. Anal.100, 13–52 (1987)Google Scholar
  9. [BJ 92]
    Ball, J.M., James, R.D.: Proposed experimental tests of a theory of fine structure and the two-well problem. Philos. Trans. R. Soc. Lond.A 338, 389–450 (1992)Google Scholar
  10. [BC 54]
    Basinski, Z., Christian, J.: Experiments on the martensitic transformation in single crystals of indium-thallium alloys. Acta Met.2, 148–166 (1954)Google Scholar
  11. [BHK 87]
    Barsch, G., Horovitz, B., Krummhansl, J.: Dynamics of twin boundaries in martensites. Phys. Rev. Lett.59, 1251–1254 (1987)Google Scholar
  12. [BMK 54]
    Bowles, J.S., MacKenzie, J.K.: The cristallography of martensitic transformations, parts I and II. Acta Met.2, 129–137, 138–147 (1954)Google Scholar
  13. [CGS 84]
    Carr, J., Gurtin, M.E., Slemrod, M.: Structured phase transitions on a finite interval. Arch. Rat. Mech. Anal.86, 317–351 (1984)Google Scholar
  14. [CP 89]
    Carr, J., Pego, R.L.: Metastable patterns in solutions of ut2uxx−f(u). Commun. Pure Appl. Math.42, 523–576 (1989)Google Scholar
  15. [DG 79]
    De Giorgi, E.: Convergence problems for functions and operators. Proc. Int. Meeting on “Recent methods in nonlinear analysis”, Rene 1978 (eds. De Giorgi E. and Magenes E.) Mosco Pitagora, Bologna, pp. 131–188Google Scholar
  16. [DD 83]
    De Giorgi, E., Dal Maso, G.: Γ-convergence and calculus of variations. In: Cecconi, J.P., Zolezzi T. (eds.) Mathematical theories of optimization (Lect. Notes Math.979, pp 121–143) Berlin Heidelberg New York: Springer 1983Google Scholar
  17. [Fo 89]
    Fonseca, I.: Phase transitions of elastic solid materials. Arch. Rat. Mech. Anal.107, 195–223 (1989)Google Scholar
  18. [FT 89]
    Fonseca, I., Tartar, L.: The gradient theory of phase transitions for systems with two potential wells. Proc. R. Soc. Edinb.111A, 89–102 (1989)Google Scholar
  19. [GH 86]
    Guckenheimer, J., Holmes, P.J.: Nonlinear oscillations, dynamical systems and bifurcation of vector fields, 2nd edn. Berlin Heidelberg New York: Springer 1986Google Scholar
  20. [HBK 91]
    Horovitz, G., Barsch G., Krummhansl, J.: Twinbands in martensites: statics and dynamics Phys. Rev.43, 1021–1033 (1991)Google Scholar
  21. [Kh 67]
    Khachaturyan, A.: Some questions concerning the theory of phase transformations in solids. Sov. Phys. -Solid State8 (9), 2163–2168 (1967)Google Scholar
  22. [Kh 83]
    Khachaturyan, A.: Theory of structural transformations in solids. New York: J. Wiley 1983Google Scholar
  23. [KS 69]
    Khachaturyan, A., Shatalov, G.: Theory of macroscopic periodicity for a phase transition in the solid state. Sov. Phys. JETP29 (3), 557–561 (1969)Google Scholar
  24. [KM 92]
    Kohn, R.V., Müller, S.: Branching of twins near a austenite/twinned-martensite interface. Philos. Mag. Ser. A66, 697–715 (1992)Google Scholar
  25. [KM 93]
    Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Commun. Pure Appl. Math. (to appear)Google Scholar
  26. [KS 89]
    Kohn, R.V., Sternberg, P.: Local minimizers and singular perturbations. Proc. R. Soc. Edinb. A111, 69–84 (1989)Google Scholar
  27. [LM 89]
    Leizarowitz, A., Mizel, V.J.: One dimensional infinite-horizon variational problems. Arch. Rat. Mech. Anal.106, 161–194 (1989)Google Scholar
  28. [Li 69]
    Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Paris: Dunod 1969Google Scholar
  29. [MH 83]
    Marsden, J.E., Hughes, T.J.R.: Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall 1983Google Scholar
  30. [Mo 87]
    Modica, L.: Gradient theory of phase transitions and minimal interface criteria. Arch. Rat. Mech. Anal.98, 357–383 (1987)Google Scholar
  31. [Mu 89]
    Müller, S.: Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations. In: Proc. “Problems involving change of type”, Stuttgart, 1988, (ed. Kirchgässner, K.) (Lect. Notes Phys.359, pp. 31–43) Berlin Heidelberg New York: Springer 1989Google Scholar
  32. [OS 91]
    Owen, N.C., Sternberg, P.: Nonconvex variational problems with anisotropic growth. Nonlin. Anal.: Theory, Methods Appl.16, 705–719 (1991)Google Scholar
  33. [RS 78]
    Reed M., Simon, B.: Methods of modern mathematical physics, vol. IV. New York: Academic Press 1978Google Scholar
  34. [Ro 69]
    Roitburd, A.: The domain structure of crystals formed in the solid phase. Sov. Phys.-Solid State10 (12), 2870–2876 (1969)Google Scholar
  35. [Ro 78]
    Roitburd, A.: Martensitic transformation as a typical phase transformation in solids. Solid State Phys.33, 317–390 (1978)Google Scholar
  36. [Wa 64]
    Wayman, C.M.: Introduction to the theory of martensitic transformations. New York: Macmillan 1964Google Scholar
  37. [WLR 53]
    Wechsler, M.S., Liebermann, D.S., Read, T.A.: On the theory of the formation of martensite. Trans. AIME J. Metals197, 1503–1515 (1953)Google Scholar
  38. [Ze 86]
    Zeidler, E.: Nonlinear functional analysis and its applications, vol. I. Berlin Heidelberg New York: Springer 1986Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Stefan Müller
    • 1
  1. 1.Institut für Angewandte MathematikUniversität BonnBonn 1Germany

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