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Γ-convergence of functionals on divergence-free fields

Published online by Cambridge University Press:  05 September 2007

Nadia Ansini  and
Adriana Garroni
Nadia Ansini
Affiliation:
Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland; nadia.ansini@epfl.ch
Adriana Garroni
Affiliation:
Dip. di Matematica, Univ. di Roma `La Sapienza', P.le A. Moro 2, 00185 Rome, Italy; garroni@mat.uniroma1.it
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Abstract

We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of Γ-convergence. We prove that the Γ-limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the Γ-limit is also stable under volume constraint and various type of boundary conditions.

Keywords

${\cal A}$-quasiconvexitydivergence-free fieldsΓ-convergencehomogenization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 4 , October 2007 , pp. 809 - 828
Copyright
© EDP Sciences, SMAI, 2007

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References

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125–145. CrossRef
R.A. Adams, Sobolev spaces. Academic Press, New York (1975).
A. Braides, Γ-convergence for Beginners. Oxford University Press, Oxford (2002).
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998).
Braides, A., Fonseca, I. and Leoni, G., A-Quasiconvexity: Relaxation and Homogenization. ESAIM: COCV 5 (2000) 539–577. CrossRef
G. Dal Maso, An Introduction to Γ -convergence. Birkhäuser, Boston (1993).
Fonseca, I. and Müller, S., A-Quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. CrossRef
Fonseca, I., Müller, S. and Pedregal, P., Analysis of concentration and oscillation effects generated by gradient. SIAM J. Math. Anal. 29 (1998) 736–756. CrossRef
Murat, F., Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1981) 68–102.
P. Pedregal, Parametrized measures and variational principles. Birkhäuser, Baston (1997).
Suquet, P., Overall potentials and extremal surfaces of power law or ideally plastic composites. J. Mech. Phys. Solids 41 (1993) 981–1002. CrossRef
Talbot, D.R.S. and Willis, J.R., Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. Roy. Soc. London A 447 (1994) 365–384. CrossRef
Talbot, D.R.S. and Willis, J.R., Upper and lower bounds for the overall properties of a nonlinear composite dielectric. II. Periodic microgeometry. Proc. Roy. Soc. London A 447 (1994) 385–396. CrossRef
Tartar, L., Compensated compactness and applications to partial differential equations. Nonlinerar Analysis and Mechanics: Heriot-Watt Symposium, R. Knops Ed., Longman, Harlow. Pitman Res. Notes Math. Ser. 39 (1979) 136–212.
R. Temam, Navier-Stokes Equations. Elsevier Science Publishers, Amsterdam (1977).