Characterization of the Multiscale Limit Associated with Bounded Sequences in BV

Rita Ferreira; Irene Fonseca

doi:10.68381/jca19023

Abstract

The notion of two-scale convergence for sequences of Radon measures with finite total variation is generalized to the case of multiple periodic length scales of oscillations. The main result concerns the characterization of

(n+1)

-scale limit pairs

(u,U)

of sequences

\{(u_\varepsilon{{\cal L}^N\!}_{\lfloor\Omega}, {Du_\varepsilon}_{\lfloor\Omega})\}_{\varepsilon>0}\subset {\cal M}(\Omega;\mathbb{R}^d)\times {\cal M}(\Omega; \mathbb{R}^{d\times N})

whenever

\{u_\varepsilon\}_{\varepsilon>0}

is a bounded sequence in

BV(\Omega;\mathbb{R}^d)

. This characterization is useful in the study of the asymptotic behavior of periodically oscillating functionals with linear growth, defined in the space

BV

of functions of bounded variation and described by

n\in\mathbb{N}

microscales, undertaken in another paper of the authors [``Reiterated homogenization in

BV

via multiscale convergence'', submitted].

Author information

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

Rita Ferreira

F.C.T./C.M.A. da U.N.L., Quinta da Torre, 2829-516 Caparica, Portugal
and: I.C.T.I. - Carnegie Mellon | Portugal

ragf@fct.unl.pt

Irene Fonseca

Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

fonseca@andrew.cmu.edu

Keywords

BV-valued measures
multiscale convergence
periodic homogenization

Mathematics Subject Classification

28B05, 26A45, 35B27

References

G. Allaire: Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992) 1482–1518.https://doi.org/10.1137/0523084 Google Scholar
G. Allaire, M. Briane: Multiscale convergence and reiterated homogenisation, Proc. R. Soc. Edinb., Sect. A 126 (1996) 297–342.https://doi.org/10.1017/S0308210500022757 Google Scholar
M. Amar: Two-scale convergence and homogenization on BV (Ω), Asymptotic Anal. 16 (1998) 65–84.https://doi.org/10.3233/ASY-1998-276 Google Scholar
L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001 Google Scholar
A. Bensoussan, J.-L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).Google Scholar
H. Brézis: Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983).Google Scholar
J. Diestel, J. J. Uhl: Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence (1977).https://doi.org/10.1090/surv/015 Google Scholar
P. Donato: Alcune osservazioni sulla convergenza debole di funzioni non uniformemente oscillanti, Ric. Mat. 32 (1983) 203–219.Google Scholar
L. Evans: Weak Convergence Methods for Nonlinear Partial Differential Equations (Chicago, 1988), CBMS 74, Amer. Math. Soc., Providence (1990).https://doi.org/10.1090/cbms/074 Google Scholar
R. Ferreira, I. Fonseca: Reiterated homogenization in BV via multiscale convergence, SIAM J. Math. Anal., to appear.Google Scholar
I. Fonseca, G. Leoni: Modern Methods in the Calculus of Variations. Lp Spaces, Springer Monographs in Mathematics, Springer, New York (2007).Google Scholar
V. Jikov, S. Kozlov, O. Oleĭnik: Homogenization of Differential Operators and Integral Functionals, Springer, Berlin (1994).https://doi.org/10.1007/978-3-642-84659-5 Google Scholar
G. Nguetseng: A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989) 608–623.https://doi.org/10.1137/0520043 Google Scholar
E. Sanchez-Palencia: Nonhomogeneous Media and Vibration Theory, Springer, Berlin (1980).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2012. Rights now held by Banach Press.

[1] G. Allaire: Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992) 1482–1518.https://doi.org/10.1137/0523084 Google Scholar

[2] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenisation, Proc. R. Soc. Edinb., Sect. A 126 (1996) 297–342.https://doi.org/10.1017/S0308210500022757 Google Scholar

[3] M. Amar: Two-scale convergence and homogenization on BV (Ω), Asymptotic Anal. 16 (1998) 65–84.https://doi.org/10.3233/ASY-1998-276 Google Scholar

[4] L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001 Google Scholar

[5] A. Bensoussan, J.-L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).Google Scholar

[6] H. Brézis: Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983).Google Scholar

[7] J. Diestel, J. J. Uhl: Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence (1977).https://doi.org/10.1090/surv/015 Google Scholar

[8] P. Donato: Alcune osservazioni sulla convergenza debole di funzioni non uniformemente oscillanti, Ric. Mat. 32 (1983) 203–219.Google Scholar

[9] L. Evans: Weak Convergence Methods for Nonlinear Partial Differential Equations (Chicago, 1988), CBMS 74, Amer. Math. Soc., Providence (1990).https://doi.org/10.1090/cbms/074 Google Scholar

[10] R. Ferreira, I. Fonseca: Reiterated homogenization in BV via multiscale convergence, SIAM J. Math. Anal., to appear.Google Scholar

[11] I. Fonseca, G. Leoni: Modern Methods in the Calculus of Variations. Lp Spaces, Springer Monographs in Mathematics, Springer, New York (2007).Google Scholar

[12] V. Jikov, S. Kozlov, O. Oleĭnik: Homogenization of Differential Operators and Integral Functionals, Springer, Berlin (1994).https://doi.org/10.1007/978-3-642-84659-5 Google Scholar

[13] G. Nguetseng: A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989) 608–623.https://doi.org/10.1137/0520043 Google Scholar

[14] E. Sanchez-Palencia: Nonhomogeneous Media and Vibration Theory, Springer, Berlin (1980).Google Scholar