Advertisement

A version of the fundamental theorem for young measures

II - Mathematical Analysis b - Miscellaneuos Topics
Part of the Lecture Notes in Physics book series (LNP, volume 344)

Keywords

Fundamental Theorem Nonlinear Partial Differential Equation Optimal Control Theory Young Measure Isometric Isomorphism 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W.K.Allard, “On the first variation of a varifold”, Annals of Math.,95 (1972) 417–491.Google Scholar
  2. [2]
    A.V.Balakrishnan, ‘Applied Functional Analysis', Springer, 1976.Google Scholar
  3. [3]
    E.J.Balder, ‘A general approach to lower semicontinuity and lower closure in optimal control theory”, SIAM J. Control and Optimization, 22 (1984) 570–598.Google Scholar
  4. [4]
    E.J.Balder, “Generalized equilibrium results for games with incomplete information”, Mathematics of Operations Research, 13 (1988) 265–276.Google Scholar
  5. [5]
    E.J.Balder, “Fatou's lemma in infinite dimensions”, J. Math. Anal. Appl., 136 (1988).Google Scholar
  6. [6]
    J.M.Ball, “Material instabilities and the calculus of variations”, Proc. conference on ‘Phase transformations and material instabilities in solids', Mathematics Research Center, University of Wisconsin, Academic Press, Publication No. 52, (1984) 1–20.Google Scholar
  7. [7]
    J.M.Ball and R D James, “Fine phase mixtures as minimizers of energy”, Arch. Rat. Mech. Anal.,100 (1987) 13–52.Google Scholar
  8. [8]
    J.M.Ball and R.D.James, “Proposed experimental tests of a theory of fine microstructure, and the two-well problem”, to appear.Google Scholar
  9. [9]
    J.M.Ball and G.Knowles, unpublished work summarised in [6].Google Scholar
  10. [10]
    J.M.Ball and F.Murat, “Remarks on Chacon's biting lemma”, to appear.Google Scholar
  11. [11]
    H.Berliocchi and J.M.Lasry, “Intégrandes normales et mesures paramétrées en calcul des variations”, Bull. Soc. Math. France, 101 (1973) 129–184.Google Scholar
  12. [12]
    I.Capuzzo Dolcetta and H.Ishii, “Approximate solutions of the Bellman equation of deterministic control theory”, Appl. Math. Optim., 11 (1984) 161–181.Google Scholar
  13. [13]
    C.Castaing and M.Valadier, ‘Convex Analysis and Measurable Multi-functions', Springer Lecture Notes in Mathematics, Vol.580, 1977.Google Scholar
  14. [14]
    M.Chipot and D.Kinderlehrer, “Equilibrium configurations of crystals”, Arch. Rat. Mech. Anal., 102 (1988) 237–278.Google Scholar
  15. [15]
    C.M.Dafermos, “Solutions in L for a conservation law with memory”, in ‘Analyse Mathématique et Applications’ Gauthier-Villars, (1988).Google Scholar
  16. [16]
    C.Dellacherie and P-A.Meyer, ‘Probabilités et Potentiel',Hermann, 1975.Google Scholar
  17. [17]
    R.J.DiPerna, “Convergence of approximate solutions to conservation laws”, Arch. Rat. Mech. Anal., 82 (1983) 27–70.Google Scholar
  18. [18]
    R.J.DiPerna, “Convergence of the viscosity method for isentropic gas dynamics”, Comm. Math. Phys. 91 (1983) 1–30.Google Scholar
  19. [19]
    R.J.DiPerna and A.J.Majda, “Oscillations and concentrations in weak solutions of the incompressible fluid equations”, Comm. Math. Phys., 108 (1987) 667–689.Google Scholar
  20. [20]
    R.J.DiPerna and A.J.Majda, “Concentrations in regularizations for 2-D incompressible flow”, Comm. Pure Appl. Math., 40 (1987) 301–345.Google Scholar
  21. [21]
    N.Dunford and J.T.Schwartz, ‘Linear Operators',Part I, Interscience, 1967.Google Scholar
  22. [22]
    R.E.Edwards, ‘Functional Analysis', Holt, Rinehart and Winston, 1965.Google Scholar
  23. [23]
    H.Federer and W.H.Fleming, “Normal and integral currents”, Annals of Math., 72 (1960) 458–520.Google Scholar
  24. [24]
    R.V.Gamkrelidze, “On sliding optimal states”, Dokl. Akad. Nauk. SSSR 143 (1962) 1243–1245 = Soviet Math. Doklady, 3 (1962) 559–561.Google Scholar
  25. [25]
    E.Hewitt and K.Stromberg, ‘Real and Abstract Analysis', Springer, 1965.Google Scholar
  26. [26]
    A.& C. Ionescu Tulcea, ‘Topics in the Theory of Lifting', Springer, New York, 1969.Google Scholar
  27. [27]
    D.Kinderlehrer, “Remarks about equilibrium configurations of crystals”, in ‘Material Instabilities in Continuum Mechanics', ed. J.M.Ball, Oxford University Press, 1988, pp.217–241.Google Scholar
  28. [28]
    E.J.MacShane, “Generalized curves”, Duke Math. J., 6 (1940) 513–536.Google Scholar
  29. [29]
    E.J.MacShane, ‘Integration', Princeton Univ. Press, 1947.Google Scholar
  30. [30]
    P-A.Meyer, ‘Probability and Potentials', Blaisdell, Waltham, 1966.Google Scholar
  31. [31]
    M.Rascle, “Un résultat de 'compacité par compensation' à coefficients variables. Application a l'elasticité non linéaire”, C.R. Acad. Sci. Paris, 302 (1986) 311–314.Google Scholar
  32. [32]
    V.Roytburd & M.Slemrod, “An application of the method of compensated compactness to a problem in phase transitions”, in ‘Material Instabilities in Continuum Mechanics', ed. J.M.Ball, Oxford University Press, 1988, pp.427–463.Google Scholar
  33. [33]
    M.E.Schonbek, “Convergence of solutions to nonlinear dispersive equations”, Comm. in Partial Diff. Equations, 7 (1982) 959–1000.Google Scholar
  34. [34]
    D.Serre, “La compacité par compensation pour les syst`emes hyperboliques non linéaires à une dimension d'espace”, J. Math. Pure et Appl., 65 (1987) 423–468.Google Scholar
  35. [35]
    M.Slemrod and V.Roytburd, “Measure-valued solutions to a problem in dynamic phase transitions”, in ‘Contemporary Mathematics', Vol.50, Amer. Math. Soc., 1987.Google Scholar
  36. [36]
    L.Tartar, “Compensated compactness and applications to partial differential equations”, in ‘Nonlinear Analysis and Mechanics', Heriot-Watt Symposium, Vol IV, Pitman Research Notes in Mathematics, 1979, pp.136–192.Google Scholar
  37. [37]
    L.Tartar, “The compensated compactness method applied to systems of conservation laws”, in ‘Systems of Nonlinear Partial Differential Equations', ed.J.M.Ball, NATO ASI Series, Vol. C111, Reidel, 1982, pp.263–285.Google Scholar
  38. [38]
    J.Warga, “Relaxed variational problems”, J. Math. Anal. Appl., 4 (1962) 111–128.Google Scholar
  39. [39]
    J.Warga, ‘Optimal Control of Differential and Functional Equations', Academic Press, 1972.Google Scholar
  40. [40]
    L.C.Young, “Generalized curves and the existence of an attained absolute minimum in the calculus of variations”, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, classe III, 30 (1937) 212–234.Google Scholar
  41. [41]
    L.C.Young, ‘Lectures on the Calculus of Variations and Optimal Control Theory', Saunders, 1969 (reprinted by Chelsea, 1980).Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  1. 1.Department of MathematicsHeriot-Watt UniversityRiccartonScotland

Personalised recommendations