We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of -convergence. We prove that the -limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the -limit is also stable under volume constraint and various type of boundary conditions.
Mots clés : ${\mathcal {A}}$-quasiconvexity, divergence-free fields, $\Gamma $-convergence, homogenization
@article{COCV_2007__13_4_809_0, author = {Ansini, Nadia and Garroni, Adriana}, title = {$\Gamma $-convergence of functionals on divergence-free fields}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {809--828}, publisher = {EDP-Sciences}, volume = {13}, number = {4}, year = {2007}, doi = {10.1051/cocv:2007041}, mrnumber = {2351405}, zbl = {1127.49011}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007041/} }
TY - JOUR AU - Ansini, Nadia AU - Garroni, Adriana TI - $\Gamma $-convergence of functionals on divergence-free fields JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 809 EP - 828 VL - 13 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007041/ DO - 10.1051/cocv:2007041 LA - en ID - COCV_2007__13_4_809_0 ER -
%0 Journal Article %A Ansini, Nadia %A Garroni, Adriana %T $\Gamma $-convergence of functionals on divergence-free fields %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 809-828 %V 13 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007041/ %R 10.1051/cocv:2007041 %G en %F COCV_2007__13_4_809_0
Ansini, Nadia; Garroni, Adriana. $\Gamma $-convergence of functionals on divergence-free fields. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 809-828. doi : 10.1051/cocv:2007041. http://www.numdam.org/articles/10.1051/cocv:2007041/
[1] Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | Zbl
and ,[2] Sobolev spaces. Academic Press, New York (1975). | MR | Zbl
,[3] -convergence for Beginners. Oxford University Press, Oxford (2002). | MR
,[4] Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). | MR | Zbl
and ,[5] A-Quasiconvexity: Relaxation and Homogenization. ESAIM: COCV 5 (2000) 539-577. | Numdam | Zbl
, and ,[6] An Introduction to -convergence. Birkhäuser, Boston (1993). | MR | Zbl
,[7] A-Quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | Zbl
and ,[8] Analysis of concentration and oscillation effects generated by gradient. SIAM J. Math. Anal. 29 (1998) 736-756. | Zbl
, and ,[9] 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. | Numdam | Zbl
,[10] Parametrized measures and variational principles. Birkhäuser, Baston (1997). | MR | Zbl
,[11] Overall potentials and extremal surfaces of power law or ideally plastic composites. J. Mech. Phys. Solids 41 (1993) 981-1002. | Zbl
,[12] 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. | Zbl
and ,[13] 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. | Zbl
and ,[14] 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. | Zbl
,[15] Navier-Stokes Equations. Elsevier Science Publishers, Amsterdam (1977).
,Cité par Sources :