On an optimal shape design problem in conduction
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 699-720.

In this paper we analyze a typical shape optimization problem in two-dimensional conductivity. We study relaxation for this problem itself. We also analyze the question of the approximation of this problem by the two-phase optimal design problems obtained when we fill out the holes that we want to design in the original problem by a very poor conductor, that we make to converge to zero.

DOI : 10.1051/cocv:2006018
Classification : 49J45, 49Q10
Mots clés : optimal shape design, relaxation, variational approach, $\Gamma $-convergence, semiconvex envelopes, quasiconvexity
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Bellido, José Carlos. On an optimal shape design problem in conduction. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 699-720. doi : 10.1051/cocv:2006018. http://www.numdam.org/articles/10.1051/cocv:2006018/

[1] G. Allaire, Shape optimization by the homogenization method. Springer (2002). | MR | Zbl

[2] G. Allaire, E. Bonnetier, G. Franfort and F. Jouve, Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27-68. | Zbl

[3] G. Allaire and R.V. Kohn, Optimal bounds on the effective behauvior of a mixture of two well-odered elastic materials. Quat. Appl. Math. 51 (1993) 643-674. | Zbl

[4] G. Allaire and R.V. Kohn, Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Europ. J. Mech. A/solids 12 (1993) 839-878. | Zbl

[5] G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated holes. Asymptotic Anal. 7 (1993) 81-95. With an appendix written jointly with A.K. Nandakumar. | Zbl

[6] J.C. Bellido, Explicit computation of the relaxed density coming from a three-dimensional optimal design prroblem. Nonlinear Analysis TMA 52 (2003) 1709-1726. | Zbl

[7] J.C. Bellido and P. Pedregal, Optimal design via variational principles: the one-dimensional case. J. Math. Pures Appl. 80 (2000) 245-261. | Zbl

[8] J.C. Bellido and P. Pedregal, Explicit quasiconvexification for some cost functionals depending on the derivatives of the state in optimal design. DCDS-A 8 (2002) 967-982. | Zbl

[9] J.C. Bellido and P. Pedregal, Optimal control via variational principles: the three dimensional case. J. Math. Anal. Appl. 287 (2003) 157-176. | Zbl

[10] J.C. Bellido and P. Pedregal, Existence in optimal control with state equation in divergence form via variational principles. J. Convex Anal. 10 (2003) 365-378. | Zbl

[11] M.P. Bendsøe and O. Sigmund, Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003). | MR | Zbl

[12] A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 22 (2002). | MR

[13] M. Briane, Homogenization in some weakly connected domains. Ricerche Mat. 47 (1998) 51-94. | Zbl

[14] M. Briane, Homogenization in general periodically perforated domains by a spectral approach. Calc. Var. Partial Differ. Equat. 15 (2002) 1-24. | Zbl

[15] A. Cherkaev, Variational methods for structural optimization. Springer (2000). | MR | Zbl

[16] G. Dal Maso, Introduction to Γ-convergence. Birkhäuser, Boston, 1993. | MR | Zbl

[17] I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition. Cal. Var. 2 (1994) 283-313. | Zbl

[18] V. Girault and P.A. Raviart, Finite elements methods for Navier-Stokes equations, Theory and Algorithms. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1985). | MR | Zbl

[19] S. Müller and V. Šverák, Convex integration for lipschitz mappings and counterexamples for regularity. Technical Report 26, Max-Planck Institute for Mathematics in the Sciences, Leipzig (1999).

[20] F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients. Ann. Mat Pura Appl. 112 (1977) 49-68. | Zbl

[21] P. Pedregal, Parametrized Measures and Variational Principles. Progress in Nonlinear Partial Differential Equations. Birkhäuser (1997). | MR | Zbl

[22] P. Pedregal, Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32 (2000) 854-869. | Zbl

[23] P. Pedregal, Constrained quasiconvexification of the square of the gradient of the state in optimal design. Quater. Appl. Math. 62 (2004) 459-470. | Zbl

[24] L. Tartar, Remarks on optimal design problems, in Homogenization and continuum mechanics, G. Buttazzo, G. Bouchitte, and P. Suchet Eds, Singapure World Scientific (1994) 279-296. | Zbl

[25] L. Tartar, An introduction to homogenization method in optimal design. Lect. Notes Math. Springer (2000). | MR | Zbl

[26] V. Šverák, Lower semicontinuity of variational integrals and compesated compactness, in Proc. ICM, S.D. Chatterji Ed., Birkhäuser 2 (1994) 1153-1158. | Zbl

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