Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T08:26:16.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Memory effect phenomena and Г-convergence

Published online by Cambridge University Press:  14 November 2011

Maria Luisa Mascarenhas
Maria Luisa Mascarenhas
Affiliation:
Centra de Matemática e Aplicaçōes Fundamentals, C.M.A.F./I.N.I.C., Av. Prof. Gama Pinto 2, 1699 Lisboa Codex, Portugal
Get access Rights & Permissions [Opens in a new window]

Synopsis

It is known that the parametric equation u'∊+ a∊u∊ = f, u∊ (0)= 0,with α ≦ a∊ ≦ β, for all ∊ > 0 and almost everywhere in a bounded domain Ω of ℝN, and f in L∞((0, T) × Ω), shows, at the limit, a memory effect. In this work the associated minimisation problem is considered and we describe how the memory effect appears in the Γ-limit, for the weak topology H1:(0, T; L2(Ω)) of the corresponding functional. The sequence a∊ has no dependence in time.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 2 , 1993 , pp. 311 - 322
Copyright
Copyright © Royal Society of Edinburgh 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Allaire, G.. Homogénéisation et convergence à deux échelles. Applications à un problème de convection diffusion. C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 581–586.Google Scholar
2Allaire, G.. Homogenization and two-scale convergence. To appear in SIAM J. Math. Anal.Google Scholar
3Ambrosio, L., P. D'Ancona and Mortola, S.. Gamma-Convergence and the Least Squares Method To appear in Ann. Mat. Pure et App.Google Scholar
4Amirat, Y., Hamdache, K. and Ziani, A.. Homogénéisation d'équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux. Ann. Inst. H. Poincaré, Anal. Nonlineaire 6 (1989), 397–417.CrossRefGoogle Scholar
5Amirat, Y., Hamdache, K. and Ziani, A.. Etude d'une équation de transport á memoire. C. R. Acad. Sci. Paris Ser. I. Math. 311 (1990), 685–688.Google Scholar
6Amirat, Y., Hamdache, K. and Ziani, A.. Homogénéisation non locale pour des équations dégénérées á coefficients périodiques. C. R. Acad. Sci. Paris Ser. I. Math. 312 (1991), 963–966.Google Scholar
7Amirat, Y., Hamdache, K. and A. Ziani. Kinetic formulation for a transport equation with memory. Comm.Partial Differential Equations 16, (1991) 1287–1311.CrossRefGoogle Scholar
8Ball, J.. A version of the fundamental Theorem for Young measures. In Partial Differential Equations and Continuum Models of Phase Transitions, Nice 1988, ed. Serre, D. (Berlin: Springer, to appear).Google Scholar
9De Giorgi, E.. Convergence Problems for Functionals and Operators. In Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Rome, May 1979, 131–188 (Bologne: Pitagora, 1979).Google Scholar
10De Giorgi, E.. Some remarks on Γ-convergence and least squares method. In Composite Media and Homogenization Theory, Trieste 1990, eds , Dall Maso and , Dell'Antonio, 135–142 (Boston: Birkhäuser, 1991).CrossRefGoogle Scholar
11Francfort, G. and Suquet, P.. Homogenization and Mechanical Dissipation in Thermoviscoelasticity. Arch. Rational Mech. Anal. 96 (1986), 265–293.CrossRefGoogle Scholar
12Mascarenhas, M. L.. A linear homogenization problem with time dependent coefficient. Trans. Amer. Math. Soc. 281 (1984), 179–195.CrossRefGoogle Scholar
13Mascarenhas, M. L.. Homogenistion of a viscoelastic equation with nonperiodic coefficients. Proc. Roy. Soc. Edinburgh Sect A 106 (1987), 143–160.CrossRefGoogle Scholar
14Mascarenhas, M. L.. Γ-limite d'une fonctionnelle lièe à un phénoméne de mémoire. C. R. Acad. Sci. Paris Ser. I. Math. (1991) 67–70.Google Scholar
15Nguetseng, G.. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623.CrossRefGoogle Scholar
16Nguetseng, G.. Asymptotic analysis for a stiff variational problem arising in Mechanics. SIAM J. Math. Anal. 21 (1990), 1394–1414.CrossRefGoogle Scholar
17Sanchez-Palencia, E.. Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127 (Berlin: Springer, 1980).Google Scholar
18Tartar, L.. Compensated compactness and applications to partial differential equations, Research Notes in Mathematics 39, Pitman Adv. Publish. Program, Heriot-Watt Symposium, vol. IV, 136–212 (London: Pitman, 1979).Google Scholar
19Tartar, L.. Memory effects and homogenization. Arch. Rational Mech. Anal. 111 (1990), 121–133.CrossRefGoogle Scholar
20Tartar, L.. Non-local effects induced by homogenization. In Partial Differential Equations and the Calculus of Variations (Essays in Honor of De Giorgi, E.) vol. II, 925–938 (Boston: Birkhäuser, 1989).Google Scholar