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Scale limit of a variational inequality modeling diffusive flux in a domain with small holes and strong adsorption in case of a critical scaling

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Abstract

In this paper we study the asymptotic behavior of solutions u ɛ of the elliptic variational inequality for the Laplace operator in domains periodically perforated by balls with radius of size C 0ɛα, C 0 > 0, α = n/n−2, and distributed with period ɛ. On the boundary of balls, we have the following nonlinear restrictions u ɛ ≥ 0, ∂ν u ɛ ≥ −ɛ−ασ(x, u ɛ), u ɛ(∂ν u ɛ + ɛ−ασ(x, u ɛ)) = 0. The weak convergence of the solutions u ɛ to the solution of an effective variational equality is proved. In this case, the effective equation contains a nonlinear term which has to be determined as solution of a functional equation. Furthermore, a corrector result with respect to the energy norm is given.

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Authors and Affiliations

  1. Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    W. Jäger & M. Neuss-Radu

  2. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia

    T. A. Shaposhnikova

Authors
  1. W. Jäger
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  2. M. Neuss-Radu
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  3. T. A. Shaposhnikova
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Correspondence to W. Jäger.

Additional information

Published in Russian in Doklady Akademii Nauk, 2011, Vol. 437, No. 4, pp. 452–456.

The article was translated by the authors.

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Jäger, W., Neuss-Radu, M. & Shaposhnikova, T.A. Scale limit of a variational inequality modeling diffusive flux in a domain with small holes and strong adsorption in case of a critical scaling. Dokl. Math. 83, 204–208 (2011). https://doi.org/10.1134/S1064562411020219

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  • DOI: https://doi.org/10.1134/S1064562411020219

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