Abstract

We obtain a version of Young's Theorem, where Young-like measures can control discontinuous functions. It determines the weak limit of {f(uν)}\{ f(u^{\nu})\} where ff is a (possibly) discontinuous scalar function, while {uν}\{u^{\nu}\} is a sequence of measurable functions which satisfies tightness condition.

Author information

Contact details are reproduced from the original publication and may be historical.

Agnieszka Kałamajska

Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland

kalamajs@mimuw.edu.pl

Keywords

  • Young measures
  • weak convergence
  • discontinuous functions

Mathematics Subject Classification

49J10, 49J45

References

  1. J. J. Alibert, G. Bouchitte: Non-uniform integrability and generalized Young measures, J. Convex Analysis 4(1) (1997) 129–147.https://doi.org/10.68381/jca04006Google Scholar
  2. J. P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  3. E. Balder: Lectures on Young measure theory and its applications to economics, Rend. Inst. Mat. Univ. Trieste 31(1) (2000) 1–69.https://doi.org/10.1090/fic/023/02Google Scholar
  4. J. M. Ball: A version of the fundamental theorem for Young measures, in: PDE’s and Continuum Models of Phase Transitions, M. Rascle, D. Serre, M. Slemrod (eds.), Lecture Notes in Physics 334, Springer, Berlin (1989) 207–215.https://doi.org/10.1007/BFb0024945Google Scholar
  5. P. G. Ciarlet: Mathematical Elasticity, Vol I: Three-Dimensional Elasticity, North-Holland, Amsterdam (1988).Google Scholar
  6. A. G. Chentsov, S. I. Morina: Extensions and Relaxations, Kluwer, Dordrecht (2002).https://doi.org/10.1007/978-94-017-1527-0Google Scholar
  7. R. J. DiPerna: Measure-valued solutions to conservation laws in continuum physics, Arch. Rational Mech. Anal. 88(3) (1985) 223–227.https://doi.org/10.1007/BF00752112Google Scholar
  8. R. J. DiPerna, A. J. Majda: Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987) 667–689.https://doi.org/10.1007/BF01214424Google Scholar
  9. L. C. Evans: Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74, AMS, Providence (1990).https://doi.org/10.1090/cbms/074Google Scholar
  10. H. O. Fattorini: Infinite-Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications 62, Cambridge University Press, Cambridge (1999).https://doi.org/10.1017/CBO9780511574795Google Scholar
  11. T. Gimse: Conservation laws with discontinuous flux functions, SIAM J. Math. Anal. 24(2) (1993) 279–289.https://doi.org/10.1137/0524018Google Scholar
  12. P. Gwiazda: An existence result for a model of granular material with non-constant density, Asympt. Anal. 30 (2002) 43–60.https://doi.org/10.3233/ASY-2002-494Google Scholar
  13. P. Gwiazda, A. Zatorska-Goldstein: An existence results for the Leray-Lions type operators with discontinuous coefficients, Asympt. Anal., to appear.Google Scholar
  14. R. E. Gladfelter, S. P. Gupta: Effect for fractional flow hysteresis on recovery of tertiary oil, SPD J. (Dec. 1980) 508–520.https://doi.org/10.2118/7577-PAGoogle Scholar
  15. J. M. N. T. Gray, M. Wieland, K. Hutter: Gravity-driven free surface flow of granular avalanches over complex basal topography, Proc. R. Soc. Lond., Ser. A 445 (1999) 1841–1874.https://doi.org/10.1098/rspa.1999.0383Google Scholar
  16. N. Hungerbühler: A refinement of Ball’s theorem on Young measures, New York J. Math. 3 (1997) 48–53.Google Scholar
  17. K. Hutter, S. Savage: The motion of a finite mass of granular material down a rough incline, J. Fluid Mech. 199 (1989) 177–215.https://doi.org/10.1017/S0022112089000340Google Scholar
  18. A. Kałamajska: Between the classical theorem of Young and convergence theorem in set-valued analysis, in preparation.Google Scholar
  19. T. Koch, R. Greve, K. Hutter: Unconfined flow of granular avalanches along a partly curved surface. II: Experiments and numerical computations, Proc. R. Soc. Lond., Ser. A 445 (1994) 399–413.https://doi.org/10.1098/rspa.1994.0068Google Scholar
  20. M. Kružík, T. Roubíček: Explicit characterization of Lp -Young measures, J. Math. Anal. Appl. 198 (1996) 830–843.https://doi.org/10.1006/jmaa.1996.0115Google Scholar
  21. M. Kružík, T. Roubíček: On the measures of Diperna and Majda, Math. Bohemica 122 (1997) 383–399.https://doi.org/10.21136/MB.1997.126212Google Scholar
  22. K. Kuratowski: Topologia, Monografie Matematyczne Tom 21, Państwowe Wydawnictwo Naukowe, Warsaw (1961), (in Polish).Google Scholar
  23. M. J. Lighthill, G. B. Whitham: On kinematic waves. I. Flood movement in long rivers, Proc. R. Soc. Lond., Ser. A 229 (1955) 281–316; II. Theory of traffic flow on long crowded roads, Proc. R. Soc. Lond., Ser. A 229 (1955) 317–345.https://doi.org/10.1098/rspa.1955.0088https://doi.org/10.1098/rspa.1955.0089Google Scholar
  24. B. B. Maini, T. Okazawa: Effects of temperature on heavy oil-vater relative permeability of sand, J. Can. Pet. Tech. 26 (1987) 33–41.https://doi.org/10.2118/87-03-03Google Scholar
  25. J. Málek, J. Necas, M. Rokyta, M. Ruzicka: Weak and Measure-Valued Solutions to Evolutionary PDE’s, Chapman & Hall, London (1996).https://doi.org/10.1007/978-1-4899-6824-1Google Scholar
  26. D. Marchesin, H. B. Medeiros, P. J. Paes-Leme: A model for two phase flow with hysteresis, Contemp. Math. 60 (1987) 89–107.https://doi.org/10.1090/conm/060/873534Google Scholar
  27. S. Migórski, A. Ochal: Optimal control of parabolic hemivariational inequalities, J. Global Optim. 17 (2000) 285–300.https://doi.org/10.1023/A:1026555014562Google Scholar
  28. S. Müller: Variational models for microstructure and phase transitions, in: Calculus of Variations and Geometric Evolution Problems, S. Hildebrandt, M. Struwe (eds.), Lectures Notes in Mathematics 1713, Springer, Berlin (1999).https://doi.org/10.1007/BFb0092670Google Scholar
  29. H. Nagahama, R. Teisseyre: Dislocation field evolution and dislocation source / sink function, Acta Geophys. Pol. 46(1) (1998) 13–33.Google Scholar
  30. Z. Naniewicz, P. D. Panagiotopoulos: Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York (1995).Google Scholar
  31. P. Pedregal: Parametrized Measures and Variational Principles, Birkhäuser, Basel (1997).https://doi.org/10.1007/978-3-0348-8886-8Google Scholar
  32. K. Pruess, Y. W. Tsang: On two-phase relative permeability and capillary pressure of rough-walled rock fractures, Water Resour. Res. 26 (1990) 1915–1926.https://doi.org/10.1029/WR026i009p01915Google Scholar
  33. Yu. G. Reshetnyak: Weak convergence of completely additive vector functions on a set, Siberian Math. J. 9 (1968) 1039–1045, Translation of Sibirsk. Mat. Z. 9 (1968) 1386–1394.https://doi.org/10.1007/BF02196453Google Scholar
  34. T. Roubíček: Relaxation in Optimalization Theory and Variational Calculus, W. de Gruyter, Berlin (1997).https://doi.org/10.1515/9783110811919Google Scholar
  35. W. Rudin: Real and Complex Analysis, McGraw-Hill, New York (1966).Google Scholar
  36. W. R. Schowalter: Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford (1978).Google Scholar
  37. K. Yosida: Functional Analysis, Springer, Berlin (1965).https://doi.org/10.1007/978-3-662-25762-3Google Scholar
  38. L. C. Young: Generalized curves and the existence of an attainment absolute minimum in the calculus of variations, C. R. Soc. Sci. Lett. Varsovie, III 30 (1937) 212–234.Google Scholar

Suggested citation

A. Kałamajska. “On Young Measures Controlling Discontinuous Functions.” Journal of Convex Analysis 13 (2006), No. 1, 177–192. https://doi.org/10.68381/jca13012

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.