Skip to main content

Advertisement

SpringerLink
Log in

Relaxation and Γ-Convergence of the Energy Functionals of a Two-Phase Elastic Medium

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The phase transition problem is studied. Some properties of the energy functionals obtained by regularization methods are established. Bibliography: 12 titles.

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. M. A. Grinfel’d, Methods of Continuum Mechanics in the Theory of Phase Transitions [in Russian], Moscow, Nauka, 1989.

    Google Scholar 

  2. V. G. Osmolovskii, Variational Problem on Phase Transitions in Mechanics of Continuous Media [in Russian], St.-Petersburg, St.-Petersburg Univ., 2000.

    Google Scholar 

  3. V. G. Osmolovskii, “Criterion for the lower semicontinuity of the energy functional of a two-phase elastic medium” [in Russian], Probl. Mat. Anal., 26 (2003), 215–254; English transl.: J. Math. Sci., 120 (2004), no. 2, 4211–4236.

    Google Scholar 

  4. A. V. Dem’yanov, “Quasiconvex pair: necessary and sufficient condition for the weak lower semi-continuity of some integral functionals” [in Russian], Probl. Mat. Anal., 29 (2004), 3–17; English transl.: J. Math. Sci., 124 (2004), no. 3, 4941–4957.

    Google Scholar 

  5. A. Dem’yanov, “Lower semicontinuity of some functionals under the PDE constraints: A-quasi-convex pair” [in Russian], Zap. Nauchn. Semin. POMI 318 (2004), 100–119.

    MathSciNet  Google Scholar 

  6. A. Braides, I. Fonseca, and G. Leoni, “A-quasiconvexity: relaxation and homogenization,” ESAIM: COCV 5 (2000), 539–577.

    Article  Google Scholar 

  7. I. Fonseca and S. Muller, “A-quasiconvexity, lower semicontinuity and Young measures,” SIAM J. Math. Anal. 30 (1999), 1355–1390.

    Article  Google Scholar 

  8. I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Amsterdam, North-Holland, 1976.

    Google Scholar 

  9. B. Dacorogna, Direct Methods in the Calculus of Variations, Berlin, Springer-Verlag, 1989.

    Google Scholar 

  10. F. Bethuel, G. Huisken, S. Muller, and K. Steken, “Variational models for microstructure and phase transitions,” in: Calculus of Variations and Geometric Evolution Problems, Lect. Notes Math. 1713 (1999), pp. 85–210.

    Google Scholar 

  11. V. G. Osmolovskii, “Exact solutions to the variational problem of the phase transition theory in continuum mechanics” [in Russian], Probl. Mat. Anal., 27 (2004), 171–205; English transl.: J. Math. Sci., 120 (2004), no. 2, 1167–1190.

    Google Scholar 

  12. J. Kristensen, “Lower semicontinuity in spaces of weakly differentiable functions,” Math. Ann. 313 (1999), 653–710.

    Article  Google Scholar 

Download references

Authors
  1. A. V. Dem’yanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 17–30.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dem’yanov, A.V. Relaxation and Γ-Convergence of the Energy Functionals of a Two-Phase Elastic Medium. J Math Sci 128, 3177–3194 (2005). https://doi.org/10.1007/s10958-005-0263-3

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-005-0263-3

Keywords

Search

Navigation