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.
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References
W.K.Allard, “On the first variation of a varifold”, Annals of Math.,95 (1972) 417–491.
A.V.Balakrishnan, ‘Applied Functional Analysis', Springer, 1976.
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.
E.J.Balder, “Generalized equilibrium results for games with incomplete information”, Mathematics of Operations Research, 13 (1988) 265–276.
E.J.Balder, “Fatou's lemma in infinite dimensions”, J. Math. Anal. Appl., 136 (1988).
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.
J.M.Ball and R D James, “Fine phase mixtures as minimizers of energy”, Arch. Rat. Mech. Anal.,100 (1987) 13–52.
J.M.Ball and R.D.James, “Proposed experimental tests of a theory of fine microstructure, and the two-well problem”, to appear.
J.M.Ball and G.Knowles, unpublished work summarised in [6].
J.M.Ball and F.Murat, “Remarks on Chacon's biting lemma”, to appear.
H.Berliocchi and J.M.Lasry, “Intégrandes normales et mesures paramétrées en calcul des variations”, Bull. Soc. Math. France, 101 (1973) 129–184.
I.Capuzzo Dolcetta and H.Ishii, “Approximate solutions of the Bellman equation of deterministic control theory”, Appl. Math. Optim., 11 (1984) 161–181.
C.Castaing and M.Valadier, ‘Convex Analysis and Measurable Multi-functions', Springer Lecture Notes in Mathematics, Vol.580, 1977.
M.Chipot and D.Kinderlehrer, “Equilibrium configurations of crystals”, Arch. Rat. Mech. Anal., 102 (1988) 237–278.
C.M.Dafermos, “Solutions in L ∞ for a conservation law with memory”, in ‘Analyse Mathématique et Applications’ Gauthier-Villars, (1988).
C.Dellacherie and P-A.Meyer, ‘Probabilités et Potentiel',Hermann, 1975.
R.J.DiPerna, “Convergence of approximate solutions to conservation laws”, Arch. Rat. Mech. Anal., 82 (1983) 27–70.
R.J.DiPerna, “Convergence of the viscosity method for isentropic gas dynamics”, Comm. Math. Phys. 91 (1983) 1–30.
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.
R.J.DiPerna and A.J.Majda, “Concentrations in regularizations for 2-D incompressible flow”, Comm. Pure Appl. Math., 40 (1987) 301–345.
N.Dunford and J.T.Schwartz, ‘Linear Operators',Part I, Interscience, 1967.
R.E.Edwards, ‘Functional Analysis', Holt, Rinehart and Winston, 1965.
H.Federer and W.H.Fleming, “Normal and integral currents”, Annals of Math., 72 (1960) 458–520.
R.V.Gamkrelidze, “On sliding optimal states”, Dokl. Akad. Nauk. SSSR 143 (1962) 1243–1245 = Soviet Math. Doklady, 3 (1962) 559–561.
E.Hewitt and K.Stromberg, ‘Real and Abstract Analysis', Springer, 1965.
A.& C. Ionescu Tulcea, ‘Topics in the Theory of Lifting', Springer, New York, 1969.
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.
E.J.MacShane, “Generalized curves”, Duke Math. J., 6 (1940) 513–536.
E.J.MacShane, ‘Integration', Princeton Univ. Press, 1947.
P-A.Meyer, ‘Probability and Potentials', Blaisdell, Waltham, 1966.
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.
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.
M.E.Schonbek, “Convergence of solutions to nonlinear dispersive equations”, Comm. in Partial Diff. Equations, 7 (1982) 959–1000.
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.
M.Slemrod and V.Roytburd, “Measure-valued solutions to a problem in dynamic phase transitions”, in ‘Contemporary Mathematics', Vol.50, Amer. Math. Soc., 1987.
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.
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.
J.Warga, “Relaxed variational problems”, J. Math. Anal. Appl., 4 (1962) 111–128.
J.Warga, ‘Optimal Control of Differential and Functional Equations', Academic Press, 1972.
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.
L.C.Young, ‘Lectures on the Calculus of Variations and Optimal Control Theory', Saunders, 1969 (reprinted by Chelsea, 1980).
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Ball, J.M. (1989). A version of the fundamental theorem for young measures. In: Rascle, M., Serre, D., Slemrod, M. (eds) PDEs and Continuum Models of Phase Transitions. Lecture Notes in Physics, vol 344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024945
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DOI: https://doi.org/10.1007/BFb0024945
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