Skip to main content
SpringerLink
Log in

Microstructures with finite surface energy: the two-well problem

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We study solutions of the two-well problem, i.e., maps which satisfy uεSO(n)A∪SO(n)B a.c. in Ω ⊂ℝn, where A and B are n×n matrices with positive determinants. This problem arises in the study of microstructure in solid-solid phase transitions. Under the additional hypothesis that the set E where the gradient lies in SO(n) A has finite perimeter, we show that u is locally only a function of one variable and that the boundary of E consists of (subsets of) hyperplanes which extend to ∂Ω and which do not intersect in Ω. This may not be the case if the assumption on E is dropped. We also discuss applications of this result to magnetostrictive materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Ambrosio, S. Mortola & V. M. Tortorelli, Functionals with linear growth defined on vector-valued BV functions, J Math. Pures Appl. 70 (1991), 269–323.

    Google Scholar 

  2. G. Anzellotti, S. Baldo & A. Visintin, Asymptotic behaviour of the Landau-Lifshitz model of ferromagnetism, Appl. Math. Opt. 23 (1991), 171–192.

    Google Scholar 

  3. P. Aviles & Y. Giga, A mathematical problem related to the physical theory of liquid crystals, Proc. Centre. Math. Analysis, Australian Nat. Univ. (eds. J. Hutchinson & L. Simon) 12 (1987), 1–16.

  4. P. Aviles & Y. Giga, Singularities and rank-1 properties of Hessian matrices, Duke Math. J. 58 (1989), 441–467.

    Google Scholar 

  5. L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, 1966.

  6. J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 337–403.

    Google Scholar 

  7. J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. R. Soc. Edinburgh A 88 (1981), 315–328.

    Google Scholar 

  8. J. M. Ball, Sets of gradients with no rank-one connection, J. Math. Pures Appl. 69 (1990), 241–259.

    Google Scholar 

  9. J. M. Ball & R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal. 100 (1987), 13–52.

    Google Scholar 

  10. J. M. Ball & R. D. James, A characterization of plane strain, Proc. R. Soc. London A 432 (1991), 93–99.

    Google Scholar 

  11. J. M. Ball & R. D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem, Phil. Trans. R. Soc. Lond. A 338 (1992), 389–450.

    Google Scholar 

  12. B. Bojarski, Generalized solutions of a system of first order differential equations of elliptic type with discontinuous coefficients (in Russian), Mat. Sb. 43 (1957), 451–503.

    Google Scholar 

  13. B. Bojarski & T. Iwaniec, Another approach to the Liouville theorem, Math. Nachr. 107 (1982), 253–262.

    Google Scholar 

  14. B. Bojarski & T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in ℝn, Ann. Acad. Sci. Fenn. Ser. A Math. 8 (1983), 257–324.

    Google Scholar 

  15. W. F. Brown, Magneto-elastic interactions, Springer, 1966.

  16. A. Cellina, On minima of a functional of the gradient: necessary conditions, Nonlin. Anal., Theory Meths. Appls. 20 (1993), 337–341.

    Google Scholar 

  17. A. Cellina, On minima of a functional of the gradient: sufficient conditions, Nonlin. Anal., Theory Meths. Appls. 20 (1993), 343–347.

    Google Scholar 

  18. M. Chipot & D. Kinderlehrer, Equilibrium configurations of crystals, Arch. Rational Mech. Anal. 103 (1988), 237–277.

    Google Scholar 

  19. A. De Simone, Energy minimizers for large ferromagnetic bodies, Arch. Rational Mech. Anal. 125 (1993), 99–143.

    Google Scholar 

  20. L. C. Evans & R. F. Gariepy, Measure theory and fine properties of functions, CRC Publ, 1992.

  21. H. Federer, Geometric measure theory, Springer, 1969.

  22. I. Fonseca, Phase transitions for elastic solid materials. Arch. Rational Mech. Anal. 107 (1989), 195–223.

    Google Scholar 

  23. I. Fonseca & W. Gangbo, Local invertibility of Sobolev functions, SIAM J. Math. Anal. 26 (1995), 280–304.

    Google Scholar 

  24. G. Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems, Proc. R. Soc. Edinburgh A 124 (1994), 437–471.

    Google Scholar 

  25. M. Giaquinta, G. Modica & J. Souček, Variational problems for maps of bounded variation with values in S1, Calc. Vor. 1 (1993), 87–121.

    Google Scholar 

  26. E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, 1984.

  27. M. Gromov, Partial differential relations, Springer, 1986.

  28. A. Hubert, Zur Theorie der zweiphasigen Domänenstrukturen in Supraleitern und Ferromagneten, Phys. Status Solidi 24 (1967), 669–682.

    Google Scholar 

  29. R. D. James & D. Kinderlehrer, Theory of diffusionless phase transitions, in: PDEs and continuum models for phase transitions (eds. M. Rascle, D. Serre & M. Slemrod), Lect. Notes Phys. 344, Springer, 1989.

  30. R. D. James & D. Kinderlehrer, Frustration in ferromagnetic materials, Cont. Mech. Thermodyn. 2 (1990), 215–239.

    Google Scholar 

  31. R. D. James & D. Kinderlehrer, Theory of magnetostriction with applications to Tb x Dy1−x Fe2, Phil. Mag. B 68 (1993), 237–274.

    Google Scholar 

  32. F. John, On quasi-isometric mappings I, Comm. Pure Appl. Math. 21 (1968), 77–110.

    Google Scholar 

  33. F. John, On quasi-isometric mappings II, Comm. Pure Appl. Math. 22 (1969), 265–278.

    Google Scholar 

  34. A. Khachaturyan, Some questions concerning the theory of phase transformations in solids, Soviet Physics-Solid State 8 (1967), 2163–2168.

    Google Scholar 

  35. A. Khachaturyan, Theory of structural transformations in solids, Wiley, 1983.

  36. A. Khachaturyan & G. Shatalov, Theory of macroscopic periodicity for a phase transition in the solid state, Soviet Physics JETP 29 (1969), 557–561.

    Google Scholar 

  37. R. V. Kohn, The relationship between linear and nonlinear variational models of coherent phase transitions, in: Trans. 7th Army Conf. on Applied Mathematics and Computing (ed. F. Dressel).

  38. R. V. Kohn & S. Müller, Branching of twins near a austenite/twinned-martensite interface, Phil. Mag. A 66 (1992), 697–715.

    Google Scholar 

  39. R. V. Kohn & S. Müller, Surface energy and microstructure in coherent phase transitions, to appear in Comm. Pure Appl. Math.

  40. R. V. Kohn & P. Sternberg, Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh A 111 (1989), 69–84.

    Google Scholar 

  41. N. H. Kuiper, On C 1 isometric embeddings, I. Proc. Konikl. Nederl. Ak. Wet. A 58 (1995), 545–556.

    Google Scholar 

  42. E. Lifshitz, On the magnetic structure of iron, J. Phys 8 (1944), 337–346.

    Google Scholar 

  43. O. Martio, S. Rickman & J. Väisälä Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 488 (1971), 1–31.

    Google Scholar 

  44. O. Martio & J. Väisälä, Elliptic equations and maps of bounded length distorsion, Math. Ann. 282 (1988), 423–443.

    Google Scholar 

  45. L. Modica, Gradient theory of phase transitions and minimal interface criteria, Arch. Rational Mech. Anal. 98 (1987), 357–383.

    Google Scholar 

  46. S. Müller, Singular perturbations as a selection criterion for periodic minimizing sequences, Calc. Var. 1 (1993), 169–204.

    Google Scholar 

  47. S. Müller & V. Šverák, Surprising attainment results for the two-well problem, preprint SFB256, Bonn University.

  48. J. Nash, C 1 isometric embeddings, Ann. Math. 60 (1954), 383–396.

    Google Scholar 

  49. I. Privorotskii, Thermodynamic theory of domain structures, Wiley, 1976.

  50. Yu. G. Reshetnyak, On the stability of conformal maps in multidimensional spaces, Sib. Math. J. 8 (1967), 69–85.

    Google Scholar 

  51. Yu. G. Reshetnyak, Liouville's theorem under minimal regularity assumptions, Sib. Math. J. 8 (1967), 631–634.

    Google Scholar 

  52. Yu. G. Reshetnyak, Space mappings with bounded distorsions, Transl. of Math. Monographs 73, Amer. Math. Soc., 1989.

  53. S. Rickman, Quasiregular mappings, Springer, 1993.

  54. A. Roitburd, The domain structure of crystals formed in the solid phase, Soviet Physics-Solid State 10 (1969), 2870–2876.

    Google Scholar 

  55. A. Roitburd, Martensitic transformation as a typical phase transformation in solids, Solid State Physics 33, Academic Press, New York, 1978, 317–390.

  56. L. Simon, Geometric measure theory, Proc. Centre for Math. Analysis 3, Australian Nat. Univ., Canberra, 1984.

  57. V. Šverák, On the problem of two wells, in: Microstructure and phase transitions, IMA Vols. Appl. Math. 54 (eds. J. Erickcsen, R. D. James, D. Kinderlehrer, & M. Luskin), Springer, 1993, pp. 183–189.

  58. V. Šverák, Quasiconvex functions with subquadratic growth, Proc. R. Soc. Lond. 433 (1991), 723–725.

    Google Scholar 

  59. A. I. Volp'ert, The spaces BV and quasilinear equations, Mat. Sb. 73 (1967), 225–267.

    Google Scholar 

  60. A. I. Vol'pert & S. I. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, M. Nijhoff, 1985.

  61. K. Zhang, Remarks on the two well problem with rotations, Proc. Amer. Math. Soc. 117 (1993), 687–688.

    Google Scholar 

  62. W. Ziemer, Weakly differentiable functions, Springer, 1989.

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Freiburg, Albertstr. 23b, D-79104, Freiburg

    Georg Dolzmann & Stefan Müller

Authors
  1. Georg Dolzmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by R. V. Kohn

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dolzmann, G., Müller, S. Microstructures with finite surface energy: the two-well problem. Arch. Rational Mech. Anal. 132, 101–141 (1995). https://doi.org/10.1007/BF00380505

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00380505

Keywords

Search

Navigation