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Young measures and the absence of fine microstructures in a class of phase transitions

Published online by Cambridge University Press:  16 July 2009

João Palhoto Matos
João Palhoto Matos
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
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Abstract

It is shown that for a class of pairs of energy wells the only Young measures having these wells as support must reduce to spatially constant Dirac masses. This implies the prediction that fine structures will be absent in certain crystal phase transitions.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 1 , March 1992 , pp. 31 - 54
Copyright
Copyright © Cambridge University Press 1992

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References

Ball, J. M. 1988a Sets of gradients with no rank-one connections. Preprint.Google Scholar
Ball, J. M. 1988b A version of the fundamental theorem for Young measures. In: Rascle, M. D., Serre, D. and Slemrod, M. (editors), PDEs and Continuum Models of Phase Transitions, Volume 344 of Lecture Notes in Physics, Springer-Verlag.Google Scholar
Ball, J. M. & James, R. D. 1987 Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., 100(1), 13–52.CrossRefGoogle Scholar
Ball, J. M. & James, R. D. 1990 Proposed experimental tests of a theory of fine microstructures and the two-well problem. Draft.Google Scholar
Berger, C., Eyraud, L., Richard, M. & Riviére, R. 1966 Étude radiocristallographique de la variation de volume pour quelques matériaux subissant des transformations de phase solide-solide. Bull. Soc. Chim. France, 628.Google Scholar
Berger, C., Richard, M. & Eyraud, L. 1965 Application de la microcalorimetrie à la determination précise des variations d'enthalpie de quelques transformations solide-solide. Bull. Soc. Chim. France, 1491.Google Scholar
Chipot, M. & Kinderlehrer, D. 1988 Equilibrium configurations of crystals. Arch. Rational Mech. Anal., 103, 237–77.CrossRefGoogle Scholar
Evans, L. C. & Gariepy, R. F. 1987 Blow up, compactness, and partial regularity in the calculus of variations. Indiana U. Math. J., 36, 2, 361–71.CrossRefGoogle Scholar
Evans, L. C. 1986 Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mccli. Anal., 95, 227–52.CrossRefGoogle Scholar
Giaquinta, M. 1983 Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Volume 105 of Annals of Mathematics Sudies, Princeton University Press.Google Scholar
Giaquinta, M. & Modica, G. 1979 Almost-everywhere regularity results for solutions of nonlinear elliptic systems. Manuscripta Math., 28, 109–58.CrossRefGoogle Scholar
Giaquinta, M. & Modica, G. 1986 Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. Henri Poincaré, Analyse non linéaire, 3, 3, 185–208.CrossRefGoogle Scholar
James, R. D. 1986 Displacive phase transformations in solids. J. Mech. Phys. Solids, 34, 359–94.CrossRefGoogle Scholar
James, R. D. 1987 The stability and metastability of quartz. In: Antman, S., Ericksen, J. L., Kinderlehrer, D. and Müller, I. (editors), Metastability and Incompletely Posed Problems. Volume 3 of IMA Vol. Math. Appl., pp. 147–76. Springer-Verlag.CrossRefGoogle Scholar
James, R. D. & Kinderlehrer, D. 1989 Theory of diffusionless phase transitions. In: Rascle, M. D., Serre, D. and Slemrod, M. (editors), PDEs and Continuum Models of Phase Transitions. Volume 344 of Lecture Notes in Physics, Springer-Verlag.Google Scholar
Kinderlehrer, D. 1988a Personal communication.Google Scholar
Kinderlehrer, D. 1988b Manuscript.Google Scholar
Kiniderlehrer, D. 1988c Remarks about equilibrium configurations of crystals. In: Ball, J. M. (editor), Symp. Material Instabilities in Continuum Mechanics, pp. 217–42, Oxford University Press.Google Scholar
Matos, J. P. 1991 Some Mathematical Methods of Mechanics. PhD thesis, University of Minnesota, Minneapolis, USA.Google Scholar
Pedregal, P. 1989 Topics in nonlinear analysis. PhD thesis, University of Minnesota, Minneapolis, USA.Google Scholar
Reshetnyak, Y. G. 1967 On the stability of conformal mappings in multidimensional spaces. Siberian Math. J., 8, 69–85.CrossRefGoogle Scholar
Schonbek, M. E. 1982 Convergence of solutions to nonlinear dispersive equations. Comm. in Partial Differential Equations, 7, 8, 959–1000.Google Scholar
Serre, D. 1983 Formes quadratiques et calcul des variations. J. Math. Pures et Appl., 62, 2, 177–196.Google Scholar
Tartar, L. 1983 The compensated compactness method applied to systems of conservation laws. In: Ball, J. M. (editor), Systems of Nonlinear Partial Differential Equations. Volume C III of NATO ASI Series, pp. 263–85, Reidel.CrossRefGoogle Scholar
Van Hove, L. 1947 Sur l'extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs functions inconnues. Proc. Koninkl. Ned. Akad. Wetenschap, 50, 18–23.Google Scholar