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Some smoothness results for the optimal design of a two-composite material which minimizes the energy

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Abstract

We study the smoothness properties of the optimal solutions corresponding to the minimization of the potential energy for the mixture of two isotropic materials (electric, thermic, elastic membrane,\(\ldots \)). As it is well known this problem has not solution in general and therefore our results refer to a relaxed formulation. We show that the state function \(u\) is twice derivable and that the optimal proportion is derivable in the direction of \(\nabla u\). We also get some uniqueness results and applications to the non-existence of classical (unrelaxed) solutions.

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Acknowledgments

This work has been partially supported by the project MTM2011-24457 of the “Ministerio de Economía y Competitividad” of Spain.

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  1. Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C. Tarfía s/n, 41012, Sevilla, Spain

    Juan Casado-Díaz

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  1. Juan Casado-Díaz
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Correspondence to Juan Casado-Díaz.

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Communicated by L. Ambrosio.

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Casado-Díaz, J. Some smoothness results for the optimal design of a two-composite material which minimizes the energy. Calc. Var. 53, 649–673 (2015). https://doi.org/10.1007/s00526-014-0762-5

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