Archive for Rational Mechanics and Analysis

, Volume 215, Issue 1, pp 1–63 | Cite as

Directional Oscillations, Concentrations, and Compensated Compactness via Microlocal Compactness Forms

  • Filip Rindler


This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in L p -bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and H-measures. In L p -spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow one to quantify the difference in compactness. The definition of MCFs involves a Fourier variable, whereby differential constraints on the functions in the sequence can also be investigated easily—a distinct advantage over Young measure theory. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar’s framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a “geometric” way. After developing several aspects of the abstract theory, we consider three applications; for lamination microstructures, the hierarchy of oscillations is reflected in the MCF. The directional information retained in an MCF is harnessed in the relaxation theory for anisotropic integral functionals. Finally, we indicate how the theory pertains to the study of propagation of singularities in certain systems of PDEs. The proofs combine measure theory, Young measures, and harmonic analysis.


Pseudodifferential Operator Young Measure Strong Compactness Differential Constraint Gradient Young Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alibert J.J., Bouchitté G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4, 129–147 (1997)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free-Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, 2000Google Scholar
  3. 3.
    Antonić, N., Mitrović, D.: H-distributions: an extension of H-measures to an L p-L q setting. Abstr. Appl. Anal. Article ID 901,084 (2011)Google Scholar
  4. 4.
    Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Coifman R.R., Rochberg R., Weiss G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, Vol. 78, 2nd edn. Springer, Berlin, 2008Google Scholar
  7. 7.
    De Lellis, C., Székelyhidi Jr., L.: The Euler equations as a differential inclusion. Ann. Math. 170, 1417–1436 (2009)Google Scholar
  8. 8.
    Diestel, J., Uhl, Jr., J.J.: Vector measures, Mathematical Surveys, Vol. 15. American Mathematical Society, Providence, 1977Google Scholar
  9. 9.
    DiPerna R.J., Majda A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108, 667–689 (1987)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dunford N.J., Schwartz J.T.: Linear Operators I: General Theory. Interscience Publishers, New York (1958)zbMATHGoogle Scholar
  11. 11.
    Fermanian Kammerer, C., Gérard, P.: A Landau–Zener formula for non-degenerated involutive codimension 3 crossings. Ann. Henri Poincaré 4, 513–552 (2003)Google Scholar
  12. 12.
    Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: L p Spaces. Springer, Berlin, 2007Google Scholar
  13. 13.
    Fonseca, I., Müller, S.: \({{\mathcal{A}}}\)-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30(6), 1355–1390 (1999)Google Scholar
  14. 14.
    Gérard, P.: Compacité par compensation et régularité deux-microlocale. In: Séminaire Équations aux Dérivées Partielles. École Polytechnique, Palaiseau, 1988–1989Google Scholar
  15. 15.
    Gérard P.: Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991)CrossRefzbMATHGoogle Scholar
  16. 16.
    Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, Vol. 249, 2nd edn. Springer, Berlin, 2008Google Scholar
  17. 17.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Grundlehren der mathematischen Wissenschaften, Vol. 256, 2nd edn. Springer, Berlin, 1990Google Scholar
  18. 18.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Grundlehren der mathematischen Wissenschaften, Vol. 274. Springer, Berlin, 2007Google Scholar
  19. 19.
    Joly, J.L., Métivier, G., Rauch, J.: Trilinear compensated compactness and nonlinear geometric optics. Ann. Math. 142, 121–169 (1995)Google Scholar
  20. 20.
    Kinderlehrer D., Pedregal P.: Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115, 329–365 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Kinderlehrer D., Pedregal P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4, 59–90 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Kirchheim, B.: Rigidity and Geometry of Microstructures. Lecture Notes 16. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 2003Google Scholar
  23. 23.
    Kristensen, J., Rindler, F.: Characterization of generalized gradient Young measures generated by sequences in W1,1 and BV. Arch. Ration. Mech. Anal. 197, 539–598 (2010) [Erratum: 203, 693–700 (2012)]Google Scholar
  24. 24.
    Kružík, M., Roubícek T.: On the measures of DiPerna and Majda. Math. Bohem. 122, 383–399 (1997)Google Scholar
  25. 25.
    McLaughlin, D., Papanicolaou, G., Tartar, L.: Weak limits of semilinear hyperbolic systems with oscillating data. Macroscopic Modelling of Turbulent Flows (Nice, 1984). Lecture Notes in Physics, Vol. 230. Springer, Berlin, 277–289, 1985Google Scholar
  26. 26.
    Mielke A.: Flow properties for Young-measure solutions of semilinear hyperbolic problems. Proc. Roy. Soc. Edinb. Sect. A 129, 85–123 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Müller S.: Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20, 1087–1095 (1999)CrossRefGoogle Scholar
  28. 28.
    Müller, S.: Variational models for microstructure and phase transitions. Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Mathematics, Vol. 1713, pp. 85–210. Springer (1999)Google Scholar
  29. 29.
    Murat F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 489–507 (1978)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Murat, F.: Compacité par compensation. II. Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978). Pitagora, 245–256, 1979Google Scholar
  31. 31.
    Murat, F.: Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8, 69–102 (1981)Google Scholar
  32. 32.
    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)Google Scholar
  33. 33.
    Rindler F.: Lower semicontinuity and Young measures in BV without Alberti’s Rank-One Theorem. Adv. Calc. Var. 5, 127–159 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Rindler, F.: A local proof for the characterization of Young measures generated by sequences in BV. J. Funct. Anal. 266, 6335–6371 (2014)Google Scholar
  35. 35.
    Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. Springer, Berlin, 2002Google Scholar
  36. 36.
    Stefanov, A.: Pseudodifferential operators with rough symbols. J. Fourier Anal. Appl. 16, 97–128 (2010)Google Scholar
  37. 37.
    Stein E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  38. 38.
    Sychev, M.A.: Characterization of homogeneous gradient Young measures in case of arbitrary integrands. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29, 531–548 (2000)Google Scholar
  39. 39.
    Székelyhidi, L., Wiedemann, E.: Young measures generated by ideal incompressible fluid flows. Arch. Ration. Mech. Anal. 206, 333–366 (2012)Google Scholar
  40. 40.
    Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV. Research Notes in Mathematics, Vol. 39. Pitman, 136–212, 1979Google Scholar
  41. 41.
    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
  42. 42.
    Tartar, L.: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinb. Sect. A 115, 193–230 (1990)Google Scholar
  43. 43.
    Tartar L.: Beyond Young measures. Meccanica 30, 505–526 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Tartar, L.: An Introduction to Navier–Stokes Equation and Oceanography. Lecture Notes of the Unione Matematica Italiana, Vol. 1. Springer, Berlin, 2006Google Scholar
  45. 45.
    Tartar, L.: The General Theory of Homogenization. A Personalized Introduction. Lecture Notes of the Unione Matematica Italiana, Vol. 7. Springer, Berlin, 2009Google Scholar
  46. 46.
    Taylor, M.E.: Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics, Vol. 100. Birkhäuser, Basel, 1991Google Scholar
  47. 47.
    Taylor, M.E.: Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, Vol. 115, 2nd edn. Springer, Berlin, 2011Google Scholar
  48. 48.
    Taylor, M.E.: Partial Differential Equations II. Qualitative Studies of Linear Equations. Applied Mathematical Sciences, Vol. 116, 2nd edn. Springer, Berlin, 2011Google Scholar
  49. 49.
    Taylor, M.E.: Partial Differential Equations III. Nonlinear Equations. Applied Mathematical Sciences, Vol. 117, 2nd edn. Springer, Berlin, 2011Google Scholar
  50. 50.
    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)zbMATHGoogle Scholar
  51. 51.
    Young L.C.: Generalized surfaces in the calculus of variations. Ann. Math. 43, 84–103 (1942)CrossRefGoogle Scholar
  52. 52.
    Young L.C.: Generalized surfaces in the calculus of variations. II. Ann. Math. 43, 530–544 (1942)zbMATHGoogle Scholar
  53. 53.
    Young, L.C.: Lectures on the Calculus of Variations and Optimal Control Theory, 2nd edn. Chelsea, New York, 1980 (Reprinted by AMS Chelsea Publishing 2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations