Skip to main content
SpringerLink
Log in

Beyond young measures

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

Young measures and their limitations are discussed. Some relations between Young measures and H-measures are described and used to analyze an example from micromagnetics. The need to improve H-measures and semi-classical measures is stressed.

Sommario

Si discutono le misure di Young e le loro limitazioni. Alcune relazioni che intercorrono tra le misure di Young e le H-misure sono descritte ed utilizzate nello studio di un esempio tratto dalla meccanica di mezzi micromagnetici. Si evidenzia in particolare la necessità di migliorare la teoria delle H-misure e di altre misure semi-classiche.

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. Tartar, L., ‘Compensated compactness and applications to partial differential equations,’ in: Knops, R. J. (Ed.),Nonlinear Analysis and Mechanics, Heriot-Watt Symposium IV, Research Notes in Mathematics, vol. 39,Pitman, London, 1979, pp. 136–212.

    Google Scholar 

  2. Tartar, L., ‘H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations,’Proc. Roy. Soc. Edinburgh, A.,115 (1990), 193–230.

    Google Scholar 

  3. Young, L. C.,Lectures on the Calculus of Variation and Optimal Control Theory, W. B. Saunders, (Philadelphia: 1969).

    Google Scholar 

  4. Berliocchi, H. and Lasry, J.-M., ‘Intégrandes normales et mesures paramétrées en calcul des variations’,Bull. Soc. Math. France,101 (1973), 129–184.

    Google Scholar 

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

    Google Scholar 

  6. Tartar, L., ‘Nonlinear constitutive relations and homogenization,’ inContemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977,North-Holland Math. Studies, vol. 30, North-Holland, Amsterdam, 1978, pp. 472–484.

    Google Scholar 

  7. DiPerna, R. J., ‘Convergence of approximate solutions to conservation laws,’Arch. Rational Mech. Anal.,82 (1983), 27–70.

    Google Scholar 

  8. Tartar, L., ‘Estimations fines de coefficients homogénéisés,’ in: Ree, P. K. (ed.),Ennio De Giorgi Colloquium, Research Notes in Mathematics, vol. 125,Pitman, London, 1985, pp. 168–187.

    Google Scholar 

  9. Francfort, G., and Murat, F., ‘Homogenization and optimal bounds in linear elasticity’,Arch. Rational Mech. Anal.,94 (1986), 307–334.

    Google Scholar 

  10. Joseph, D.,Stability of Fluid Motions, I, II, Springer Tracts in Natural Philosophy, vol. 27, 28, Springer, 1976.

  11. Murat, F. and Tartar, L.C., ‘Calcul des variations et homogénéisation,’Les Méthodes de l'Homogénéisation: Théorie et Applications en Physique, Coll. de la Dir. des Etudes et Recherches de Elec. de France,57,Eyrolles,Paris, 1985, pp. 319–369.

    Google Scholar 

  12. Ball, J. M., ‘A version of the fundamental theorem for Young measures’, in: Rascle, M. Serre, D. and Slemrod, M. (Eds.),P.D.E's and Continuum Models of Phase Transitions, Lect. Notes in Phys., vol. 344, Springer, 1989, pp. 207–215.

  13. DiPerna, R. J., and Majda, A., ‘Oscillations and concentration in weak solutions of the incompressible fluid equations,’Comm. Math. Phys.,108 (1987), 667–689.

    Google Scholar 

  14. Brown, W. F.,Micromagnetics, Interscience, New York, 1963.

    Google Scholar 

  15. James, R. D. and Kinderlehrer, D., ‘Frustration in ferromagnetic materials,’Cont. Mech. Therm.,2 (1990), 215–239.

    Google Scholar 

  16. De Simone, A.,Magnetization curves in ferromagnetic materials, Ph.D. thesis, University of Minnesota, 1992.

  17. Murat, F., and Tartar, L., ‘On the relation between Young measures and H-measures,’ in preparation.

  18. Gérard, P., ‘Microlocal defect measures,’Comm. Partial Differential Equations,16 (11) (1991), 1761–1794.

    Google Scholar 

  19. Gérard, P., ‘Mesures semi-classiques et ondes de Bloch,’ inEquations aux Dérivées Partielles, Exposé XVI, Séminaire 1990–1991,Ecole Polytechnique, Palaiseau, pp. XV-1–XVI-18.

    Google Scholar 

  20. Lions, P.-L. and Paul, T., ‘Sur les mesures de Wigner,’Revista Matemática Iberoamericana,9 (1993), 261–270.

    Google Scholar 

  21. Schwartz L.,Théorie des distributions, Hermann, Paris, 1966.

    Google Scholar 

  22. Müller, S., ‘Mimimizing sequences for nonconvex functionals, phase transitions and singular perturbations’, in: Kirchgässner K. (ed.),Problems Involving Change of Type, Lect. Notes in Phys., 359, Springer-Verlag, 1990, pp. 31–44.

  23. Tartar, L.,Homogenization, Compensated Compactness and H-Measures, CBMS-NSF Conference, Santa Cruz, June 1993, (Lectures notes in preparation).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Carnegie-Mellon University, 15213-3890, Pittsburgh, PA, USA

    Luc Tartar

Authors
  1. Luc Tartar
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tartar, L. Beyond young measures. Meccanica 30, 505–526 (1995). https://doi.org/10.1007/BF01557082

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01557082

Key words

Search

Navigation