Abstract
From the extensions enumerated in Sect. 1.2, we would like to examine two important ones. Some of them are so hard to deal with that they are fully open problems, and whatever new development would have a tremendous impact. For some others, only partial (or even very partial) results are known. Yet for some other situations, extension is almost immediate. The significance of non-linear situations (either in the cost functional or in the state law) for Engineering is not covered here, and should be explored in the bibliography provided. Our motivation is purely analytical.
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References
Aranda, E., Pedregal, P.: Constrained envelope for a general class of design problems. Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Contin. Dyn. Syst. suppl, 30–41 (2003)
Boussaid, O., Pedregal, P.: Quasiconvexification of sets in optimal design. Calc. Var. Partial Differential Equations 34, 139–152 (2009)
Casado-Díaz, J., Couce-Calvo, J., Luna-Laynez, M., Martín-Gómez, J.D.: Optimal design problems for a non-linear cost in the gradient: numerical results. Appl. Anal. 87, 1461–1487 (2008)
Donoso, A.: Optimal design modeled by Poisson’s equation in the presence of gradients in the objective. Ph.D. Thesis, Universidad de Castilla-La Mancha (2004)
Donoso, A., Pedregal, P.: Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field. Struct. Multidiscip. Optim. 30, 360–367 (2005)
Fidalgo-Prieto, U., Pedregal, P.: A general lower bound for the relaxation of an optimal design problem with a general quadratic cost functional, and a general linear state equation. J. Convex Anal. 19, 281–294 (2012)
Grabovsky, Y.: Homogenization in an optimal design problem with quadratic weakly discontinuous objective functional. Int. J. Differ. Equ. Appl. 3, 183–194 (2001)
Grabovsky, Y.: Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals. Adv. Appl. Math. 27, 683–704 (2001)
Pedregal, P.: Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design. Electron. Res. Announc. Am. Math. Soc. 7, 72–78 (2001)
Pedregal, P.: Vector variational problems and applications to optimal design. ESAIM Control Optim. Calc. Var. 11, 357–381 (2005)
Pedregal, P.: Optimal design in two-dimensional conductivity for a general cost depending on the field. Arch. Ration. Mech. Anal. 182, 367–385 (2006)
Pedregal, P.: Weak limits in nonlinear conductivity. SIAM J. Math. Anal. 47 (1), 1154–1168 (2015)
Pedregal, P., Zhang, Y.: Optimal design for multimaterials. Anal. Appl. 10, 413–438 (2012)
Tartar, L.: Remarks on optimal design problems. In: Calculus of Variations, Homogenization and Continuum Mechanics (Marseille, 1993). Advances in Applied Mathematics, vol. 18, pp. 279–296. World Scientific, River Edge (1994)
Tartar, L.: Remarks on the homogenization method in optimal design methods. In: Homogenization and applications to material sciences (Nice, 1995). GAKUTO International Series. Mathematical Sciences and Applications, vol. 9, pp. 393–412. Gakktosho, Tokyo (1995)
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Pedregal, P. (2016). Some Extensions. In: Optimal Design through the Sub-Relaxation Method. SEMA SIMAI Springer Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-41159-0_6
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