Asymptotic behaviour of equicoercive diffusion energies in dimension two

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Abstract

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F n , \({n\in\mathbb{N}}\) , defined in L 2(Ω), for a bounded open subset Ω of \({\mathbb{R}^2}\) . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.

Keywords

35B27 35J20 31C25 
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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Centre de MathématiquesI.N.S.A. de Rennes & I.R.M.A.R.Rennes CedexFrance
  2. 2.Dpto. de Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain

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