Archive for Rational Mechanics and Analysis

, Volume 224, Issue 3, pp 1087–1125 | Cite as

Characterization of Generalized Young Measures Generated by Symmetric Gradients

  • Guido De Philippis
  • Filip Rindler
Open Access


This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer–Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The “local” proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti’s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences.


  1. 1.
    Alberti G.: Rank one property for derivatives of functions with bounded variation. Proc. R. Soc. Edinburgh Sect. A 123, 239–274 (1993)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alibert J.J., Bouchitté G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4, 129–147 (1997)MathSciNetMATHGoogle Scholar
  3. 3.
    Ambrosio L., Coscia A., Dal Maso G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139, 201–238 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free-Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, 2000Google Scholar
  5. 5.
    Arroyo-Rabasa, A., De Philippis, G., Rindler, F.: Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints, arXiv:1701.02230
  6. 6.
    Babadjian J.-F.: Traces of functions of bounded deformation. Indiana Univ. Math. J. 64, 1271–1290 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baia M., Matias J., Santos P.: Characterization of generalized Young measures in the \({\mathcal{A}}\)-quasiconvexity context. Indiana Univ. Math. J. 62, 487–521 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ball, J.M.: A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice, 1988), Lecture Notes in Physics, vol. 344, Springer, pp. 207–215, 1989Google Scholar
  9. 9.
    Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Barroso A.C., Fonseca I., Toader R.: A relaxation theorem in the space of functions of bounded deformation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29, 19–49 (2000)MathSciNetMATHGoogle Scholar
  11. 11.
    Benešová, B., Kružík, M.: Characterization of gradient Young measures generated by homeomorphisms in the plane. ESAIM Control Optim. Calc. Var. 22, 267–288 (2016)Google Scholar
  12. 12.
    Conti S., Faraco D., Maggi F.: A new approach to counterexamples to L 1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions. Arch. Ration. Mech. Anal. 175, 287–300 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Conti, S., Focardi, M., Iurlano, F.: Which special functions of bounded deformation have bounded variation? Proc. Roy. Soc. Edinburgh Sect. A. arXiv:1502.07464 (2015)
  14. 14.
    Conway, J.B.: A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer, 1990Google Scholar
  15. 15.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd ed., Applied Mathematical Sciences, vol. 78, Springer, 2008Google Scholar
  16. 16.
    De Philippis, G., Rindler, F.: On the structure of \({\mathcal{A}}\)-free measures and applications. Ann. Math. 184, 1017–1039 (2016)Google Scholar
  17. 17.
    DiPerna R.J., Majda A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108, 667–689 (1987)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ebobisse F.: On lower semicontinuity of integral functionals in \({LD(\Omega)}\). Ricerche Mat. 49, 65–76 (2000)MathSciNetMATHGoogle Scholar
  19. 19.
    Fonseca I., Kružík M.: Oscillations and concentrations generated by \({\mathcal{A}}\)-free mappings and weak lower semicontinuity of integral functionals. ESAIM Control Optim. Calc. Var. 16, 472–502 (2010)MATHGoogle Scholar
  20. 20.
    Fonseca I., Müller S.: \({\mathcal{A}}\)-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30, 1355–1390 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Fonseca I., Müller S., Pedregal P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kałamajska A., Kružík M.: Oscillations and concentrations in sequences of gradients. ESAIM Control Optim. Calc. Var. 14, 71–104 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kinderlehrer D., Pedregal P.: Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115, 329–365 (1991)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kinderlehrer D., Pedregal P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4, 59–90 (1994)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kirchheim B., Kristensen J.: On rank-one convex functions that are homogeneous of degree one. Arch. Ration. Mech. Anal. 221, 527–558 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kohn R. V.: New integral estimates for deformations in terms of their nonlinear strains. Arch. Ration. Mech. Anal. 78, 131–172 (1982)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kristensen J.: Lower semicontinuity in spaces of weakly differentiable functions. Math. Ann. 313, 653–710 (1999)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kristensen, J., Rindler, F.: Characterization of generalized gradient Young measures generated by sequences in \({{{\rm W}^{1,1}}}\) and BV. Arch. Ration. Mech. Anal. 197, 539–598 (2010), Erratum: Vol. 203, 693–700 (2012)Google Scholar
  29. 29.
    Kružík, M., Roubíček, T.: Explicit characterization of L p-Young measures. J. Math. Anal. Appl. 198, 830–843 (1996)Google Scholar
  30. 30.
    Kružík M., Roubíček T.: On the measures of DiPerna and Majda. Math. Bohem. 122, 383–399 (1997)MathSciNetMATHGoogle Scholar
  31. 31.
    Massaccesi, A., Vittone, D.: An elementary proof of the rank one theorem for BV functions, To appear on JEMS, 2016Google Scholar
  32. 32.
    Matthies, H., Strang, G., Christiansen, E.: The saddle point of a differential program, Energy methods in finite element analysis, Wiley, pp. 309–318, 1979Google Scholar
  33. 33.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, 1995Google Scholar
  34. 34.
    Morrey C.B. Jr.: Quasiconvexity and the semicontinuity of multiple integrals. Pacif. J. Math. 2, 25–53 (1952)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ornstein D.: A non-inequality for differential operators in the L 1 norm. Arch. Ration. Mech. Anal. 11, 40–49 (1962)CrossRefMATHGoogle Scholar
  36. 36.
    Pedregal, P.: Parametrized Measures and Variational Principles, Progress in Nonlinear Differential Equations and their Applications, vol. 30, Birkhäuser, 1997Google Scholar
  37. 37.
    Rindler, F.: Lower Semicontinuity and Young Measures for Integral Functionals with Linear Growth, Ph.D. thesis, University of Oxford, 2011Google Scholar
  38. 38.
    Rindler F.: Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures. Arch. Ration. Mech. Anal. 202, 63–113 (2011)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Rindler F.: Lower semicontinuity and Young measures in BV without Alberti’s Rank-One Theorem. Adv. Calc. Var. 5, 127–159 (2012)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Rindler F.: A local proof for the characterization of Young measures generated by sequences in BV. J. Funct. Anal. 266, 6335–6371 (2014)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Suquet P.-M.: Existence et régularité des solutions des équations de la plasticité. C. R. Acad. Sci. Paris Sér. A 286, 1201–1204 (1978)MathSciNetMATHGoogle Scholar
  42. 42.
    Suquet P.-M.: Un espace fonctionnel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. 1, 77–87 (1979)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Sychev M.A.: A new approach to Young measure theory, relaxation and convergence in energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 773–812 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Tartar, L.: Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, pp. 136–212, 1979Google Scholar
  45. 45.
    Tartar, L.: The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, pp. 263–285, 1983Google Scholar
  46. 46.
    Temam, R.: Mathematical Problems in Plasticity, Gauthier-Villars, 1985Google Scholar
  47. 47.
    Temam R., Strang G.: Functions of bounded deformation. Arch. Ration. Mech. Anal. 75, 7–21 (1980)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Young L. C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. C. R. Soc. Sci. Lett. Varsovie, Cl. III 30, 212–234 (1937)MATHGoogle Scholar
  49. 49.
    Young L. C.: Generalized surfaces in the calculus of variations. Ann. of Math. 43, 84–103 (1942)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Young L. C.: Generalized surfaces in the calculus of variations. II. Ann. of Math. 43, 530–544 (1942)MathSciNetCrossRefMATHGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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