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Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV

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An Erratum to this article was published on 22 December 2011

Abstract

Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys 108:667–689, 1987) provide a quantitative tool for studying the one-point statistics of oscillation and concentration in sequences of functions. In this work, after developing a functional-analytic framework for such measures, including a compactness theorem and results on the generation of such Young measures by L1-bounded sequences (or even by sequences of bounded Radon measures), we turn to investigation of those Young measures that are generated by bounded sequences of W1,1-gradients or BV-derivatives. We provide several techniques to manipulate such measures (including shifting, averaging and approximation by piecewise-homogeneous Young measures) and then establish the main new result of this work, the duality characterization of the set of (BV- or W1,1-)gradient Young measures in terms of Jensen-type inequalities for quasiconvex functions with linear growth at infinity. This result is the natural generalization of the Kinderlehrer–Pedregal Theorem (Arch Ration Mech Anal 115:329–365, 1991; J Geom Anal 4:59–90, 1994) for classical Young measures to the W1,1- and BV-case and contains its version for weakly converging sequences in W1,1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.

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References

  1. Acerbi, E., Fusco, N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86, 125–145 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alberti, G.: A Lusin type theorem for gradients. J. Funct. Anal. 100, 110–118 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alberti, G.: Rank one property for derivatives of functions with bounded variation. Proc. R. Soc. Edinb. Sect. A 123, 239–274 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Alibert, J.J., Bouchitté, G.: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4, 129–147 (1997)

    MATH  MathSciNet  Google Scholar 

  5. Amar, M., De Cicco, V., Fusco, N.: A relaxation result in BV for integral functionals with discontinuous integrands. ESAIM Control Optim. Calc. Var. 13, 396–412 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Amar, M., De Cicco, V., Fusco, N.: Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands. ESAIM Control Optim. Calc. Var. 14, 456–477 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ambrosio, L., Dal Maso, G.: On the relaxation in BV(Ω;R m) of quasi-convex integrals. J. Funct. Anal. 109, 76–97 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free-Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000

  9. Balder, E.J.: A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22, 570–598 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Balder, E.J.: A general denseness result for relaxed control theory. Bull. Austral. Math. Soc. 30, 463–475 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  11. Balder, E.J.: Consequences of denseness of Dirac Young measures. J. Math. Anal. Appl. 207, 536–540 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Balder, E.J.: New fundamentals of Young measure convergence. In: Calculus of Variations and Optimal Control (Haifa, 1998), Chapman & Hall/CRC Res. Notes Math., vol. 411, pp. 24–48. Chapman & Hall/CRC, Boca Raton, 2000

  13. Ball, J.M.: A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions (Nice, 1988). Lecture Notes in Physics, vol. 344, pp. 207–215. Springer, Berlin, 1989

  14. Ball, J.M., Kirchheim, B., Kristensen, J.: Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11, 333–359 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ball, J.M., Murat, F.: W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58, 225–253 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ball, J.M., Murat, F.: Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107, 655–663 (1989)

    MATH  MathSciNet  Google Scholar 

  17. Berliocchi, H., Lasry, J.M.: Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101, 129–184 (1973)

    MATH  MathSciNet  Google Scholar 

  18. Bildhauer, M.: Convex variational problems. In: Lecture Notes in Mathematics, vol. 1818. Springer, Berlin, 2003

  19. Bouchitté, G., Fonseca, I., Mascarenhas, L.: A global method for relaxation. Arch. Ration. Mech. Anal. 145, 51–98 (1998)

    Article  MATH  Google Scholar 

  20. Castaing, C., de Fitte, P.R., Valadier, M.: Young Measures on Topological Spaces: With Applications in Control Theory and Probability Theory. Mathematics and Its Applications. Kluwer, Dordrecht, 2004

  21. Chandler, R.E.: Hausdorff compactifications. In: Lecture Notes in Pure and Applied Mathematics, vol. 23. Marcel Dekker, New York, 1976

  22. Conway, J.B.: A course in functional analysis. In: Graduate Texts in Mathematics, vol. 96, 2nd edn. Springer, Berlin, 1990

  23. Dacorogna, B.: Direct methods in the calculus of variations. In: Applied Mathematical Sciences, vol. 78, 2nd edn. Springer, Berlin, 2008

  24. Dal Maso, G., DeSimone, A., Mora, M.G., Morini, M.: Time-dependent systems of generalized Young measures. Netw. Heterog. Media 2, 1–36 (2007)

    MATH  MathSciNet  Google Scholar 

  25. De Cicco, V., Fusco, N., Verde, A.: On L 1-lower semicontinuity in BV. J. Convex Anal. 12, 173–185 (2005)

    MATH  MathSciNet  Google Scholar 

  26. de Guzmán, M.: Differentiation of integrals in R n. In: Lecture Notes in Mathematics, vol. 481. Springer, Berlin, 1975

  27. Delladio, S.: Lower semicontinuity and continuity of functions of measures with respect to the strict convergence. Proc. R. Soc. Edinb. Sect. A 119, 265–278 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  28. Diestel, J., Uhl, Jr., J.J.: Vector measures. In: Mathematical Surveys, vol. 15. American Mathematical Society, Providence, 1977

  29. Dinculeanu, N.: Vector measures. In: International Series of Monographs in Pure and Applied Mathematics, vol. 95. Pergamon Press, New York, 1967

  30. DiPerna, R.J., Majda, A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108, 667–689 (1987)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Dudley, R.M.: Real analysis and probability. In: Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge, 2002

  32. Edwards, R.E.: Functional Analysis. Theory and Applications. Holt (Rinehart and Winston), New York, 1965

  33. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1992

  34. Folland, G.: Real analysis. In: Pure and Applied Mathematics, 2nd edn. Wiley, New York, 1999

  35. Fonseca, I., Müller, S.: Relaxation of quasiconvex functionals in BV(Ω,R p) for integrands f(x, u, ∇ u). Arch. Ration. Mech. Anal. 123, 1–49 (1993)

    Article  MATH  Google Scholar 

  36. Fonseca, I., Müller, S., Pedregal, P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)

    MATH  Google Scholar 

  37. Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. I, Ergebnisse der Mathematik und ihrer Grenzgebiete vol. 37. Springer, Berlin, 1998

  38. Hahn, H.: Über Annäherungen an Lebesgue’sche Integrale durch Riemann’sche Summen. Sitzungsber. Math. Phys. Kl. K. Akad. Wiss. Wien 123, 713–743 (1914)

    Google Scholar 

  39. Kałamajska, A.: On Young measures controlling discontinuous functions. J. Convex Anal. 13, 177–192 (2006)

    MATH  MathSciNet  Google Scholar 

  40. Kałamajska, A., Kružík, M.: Oscillations and concentrations in sequences of gradients. ESAIM Control Optim. Calc. Var. 14, 71–104 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Kinderlehrer, D., Pedregal, P.: Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115(4), 329–365 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. Kinderlehrer, D., Pedregal, P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4, 59–90 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  43. Kristensen, J.: Finite functionals and Young measures generated by gradients of Sobolev functions. MAT-Report 1994–34, Math. Inst., Technical University of Denmark, 1994

  44. Kristensen, J.: Lower semicontinuity of quasi-convex integrals in BV. Calc. Var. Partial Differ. Equ. 7(3), 249–261 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  46. Kristensen, J., Rindler, F.: Relaxation of signed integral functionals in BV. Calc. Var. Partial Differ. Equ. (2008, in press)

  47. Kružík, M., Roubíček, T.: On the measures of DiPerna and Majda. Math. Bohem. 122, 383–399 (1997)

    MATH  MathSciNet  Google Scholar 

  48. Morrey, Jr., C.B.: Multiple integrals in the calculus of variations. In: Grundlehren der Mathematischen Wissenschaften, vol. 130. Springer, Berlin, 1966

  49. Müller, S.: On quasiconvex functions which are homogeneous of degree 1. Indiana Univ. Math. J. 41, 295–301 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  50. Müller, S.: Variational models for microstructure and phase transitions. In: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). Lecture Notes in Mathematics, vol. 1713, pp. 85–210. Springer, Berlin, 1999

  51. Pedregal, P.: Parametrized measures and variational principles. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 30. Birkhäuser, Basel, 1997

  52. Reshetnyak, Y.G.: Weak convergence of completely additive vector functions on a set. Sib. Math. J. 9, 1039–1045 (1968)

    MATH  Google Scholar 

  53. Roubíček, T.: Relaxation in optimization theory and variational calculus. In: De Gruyter Series in Nonlinear Analysis & Applications, vol. 4. De Gruyter, New York, 1997

  54. Rudin, W.: Real and Complex Analysis 3rd edn. McGraw-Hill, New York, 1986

    Google Scholar 

  55. 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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  56. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, vol. IV. Res. Notes in Math., vol. 39, pp. 136–212. Pitman, London, 1979

  57. 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)

    MATH  Google Scholar 

  58. Young, L.C.: Generalized surfaces in the calculus of variations. Ann. Math. 43, 84–103 (1942)

    Article  Google Scholar 

  59. Young, L.C.: Generalized surfaces in the calculus of variations. II. Ann. Math. 43, 530–544 (1942)

    MATH  Google Scholar 

  60. Ziemer, W.P.: Weakly differentiable functions. In: Graduate Texts in Mathematics, vol. 120. Springer, Berlin, 1989

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  1. Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, UK

    Jan Kristensen & Filip Rindler

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  1. Jan Kristensen
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Communicated by G. Dal Maso

An erratum to this article can be found online at http://dx.doi.org/10.1007/s00205-011-0477-0

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Kristensen, J., Rindler, F. Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV. Arch Rational Mech Anal 197, 539–598 (2010). https://doi.org/10.1007/s00205-009-0287-9

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