We consider the family of materials obtained, via homogenization, by replacing a small portion, of size ɛ, of a fixed material by other materials. In a previous paper we have obtained a subset of the set of ‘derivatives’ of this family with respect to ɛ in . In the present Note we prove that this set is, in fact, dense. This result can be applied, for example, to obtain optimality conditions for composite materials.
On considère une famille de matériaux obtenus par homogénéisation consistant à remplacer une petite partie de matériau, de taille ɛ, par d'autres matériaux. Dans un article antérieur on a caractérisé un sous-ensemble de l'ensemble des « dérivées », par rapport à ɛ de cette famille, pour . Dans cette Note on démontre que ce sous-ensemble est en fait dense. Le résultat peut être appliqué, par exemple, à l'obtention des conditions d'optimalité pour des matériaux composites.
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Juan Casado-Díaz 1; Julio Couce-Calvo 1; José Domingo Martín-Gómez 1
@article{CRMATH_2006__342_5_353_0, author = {Juan Casado-D{\'\i}az and Julio Couce-Calvo and Jos\'e Domingo Mart{\'\i}n-G\'omez}, title = {A density result for the variation of a material with respect to small inclusions}, journal = {Comptes Rendus. Math\'ematique}, pages = {353--358}, publisher = {Elsevier}, volume = {342}, number = {5}, year = {2006}, doi = {10.1016/j.crma.2005.12.021}, language = {en}, }
TY - JOUR AU - Juan Casado-Díaz AU - Julio Couce-Calvo AU - José Domingo Martín-Gómez TI - A density result for the variation of a material with respect to small inclusions JO - Comptes Rendus. Mathématique PY - 2006 SP - 353 EP - 358 VL - 342 IS - 5 PB - Elsevier DO - 10.1016/j.crma.2005.12.021 LA - en ID - CRMATH_2006__342_5_353_0 ER -
%0 Journal Article %A Juan Casado-Díaz %A Julio Couce-Calvo %A José Domingo Martín-Gómez %T A density result for the variation of a material with respect to small inclusions %J Comptes Rendus. Mathématique %D 2006 %P 353-358 %V 342 %N 5 %I Elsevier %R 10.1016/j.crma.2005.12.021 %G en %F CRMATH_2006__342_5_353_0
Juan Casado-Díaz; Julio Couce-Calvo; José Domingo Martín-Gómez. A density result for the variation of a material with respect to small inclusions. Comptes Rendus. Mathématique, Volume 342 (2006) no. 5, pp. 353-358. doi : 10.1016/j.crma.2005.12.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.12.021/
[1] Shape Optimization by the Homogenization Method, Appl. Math. Sci., vol. 146, Springer-Verlag, New York, 2002
[2] Optimality conditions for nonconvex multistate control problems in the coefficients, SIAM J. Control Optim., Volume 43 (2004) no. 1, pp. 216-239
[3] Variational Methods for Structural Optimization, Appl. Math. Sci., vol. 140, Springer-Verlag, New York, 2000
[4] G. Dal Maso, R.V. Kohn, The local character of G-closure, Unpublished work
[5] Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994
[6] An -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 3 (1963) no. 17, pp. 189-206
[7] H-convergence (A. Cherkaev; R. Kohn, eds.), Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1997, pp. 21-44
[8] Calculus of variations (A. Cherkaev; R. Kohn, eds.), Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1997, pp. 139-173
[9] Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1984
[10] An Introduction to the homogenization method in optimal design, CIM/CIME, Summer School, Trôia, 1–6 June 1998 (A. Cellina; A. Ornelas, eds.) (Lecture Notes in Math.), Volume vol. 1740, Springer-Verlag, Berlin (2000), pp. 47-156
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