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The Effectiveness Factor of Reaction-Diffusion Equations: Homogenization and Existence of Optimal Pellet Shapes

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Abstract

We study the asymptotic behaviour of the so-called effectiveness factor ηε of a nonlinear diffusion equation that occurs on the boundary of periodically distributed inclusions (or particles) in an ε-periodic medium. Here, ε is a small parameter related to the characteristic size of the inclusions, which, in the homogenization process, will tend to 0. The inclusions are modeled as homothecy of a fixed pellet T, rescaled by a factor r(ε). We study the cases in which r(ε) = O(εα), known as big holes, for α = 1, as well as non-critical small holes, for

$$1 > \alpha \frac{n}{{n - 2}}$$

. We will prove the existence of some convex shapes which maximize the effectiveness of the homogenized problem. In particular, we deduce that for small holes the sphere is the domain of highest effectiveness.

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Authors and Affiliations

  1. Dpto. Matemática Aplicada & Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Spain

    Jesús Ildefonso Díaz & David Gómez-Castro

  2. Faculty of Physics, University of Bucharest, Romania

    Claudia Timofte

Authors
  1. Jesús Ildefonso Díaz
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  2. David Gómez-Castro
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  3. Claudia Timofte
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Correspondence to Jesús Ildefonso Díaz.

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Dedicated to an exceptional mathematician, David Kinderlehrer, with admiration.

The research of D. Gómez-Castro is supported by a FPU fellowship from the Spanish government. The research of J.I. Díaz and D. Gómez-Castro was partially supported by the project ref. MTM 2014-57113-P of the DGISPI (Spain).

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Díaz, J.I., Gómez-Castro, D. & Timofte, C. The Effectiveness Factor of Reaction-Diffusion Equations: Homogenization and Existence of Optimal Pellet Shapes. J Elliptic Parabol Equ 2, 119–129 (2016). https://doi.org/10.1007/BF03377396

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