Change of homogenized absorption term in diffusion processes with reaction on the boundary of periodically distributed asymmetric particles of critical size

Date

2017-07-13

Authors

Diaz, Jesus Ildefonso
Gomez-Castro, David
Shaposhnikova, Tatiana A.
Zubova, Maria

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Journal ISSN

Volume Title

Publisher

Texas State University, Department of Mathematics

Abstract

The main objective of this article is to get a complete characterization of the homogenized global absorption term, and to give a rigorous proof of the convergence, in a class of diffusion processes with a reaction on the boundary of periodically "microscopic" distributed particles (or holes) given through a nonlinear microscopic reaction (i.e. under nonlinear Robin microscopic boundary conditions). We introduce new techniques to deal with the case of non necessarily symmetric particles (or holes) of critical size which leads to important changes in the qualitative global homogenized reaction (such as it happens in many problems of the Nanotechnology). Here we shall merely assume that the particles (or holes) Gjε, in the n-dimensional space, are diffeomorphic to a ball (of diameter αε = C0εγ, γ = n/n-2 for some C0 > 0). To define the corresponding "new strange term" we introduce a one-parametric family of auxiliary external problems associated to canonical cellular problem associated to the prescribed asymmetric geometry G0 and the nonlinear microscopic boundary reaction σ(s) (which is assumed to be merely a Hölder continuous function). We construct the limit homogenized problem and prove that it is a well-posed global problem, showing also the rigorous convergence of solutions, as ε → 0, in suitable functional spaces. This improves many previous papers in the literature dealing with symmetric particles of critical size.

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Keywords

Homogenization, Diffusion processes, Periodic asymmetric particles, Microscopic non-linear boundary reaction, Critical sizes

Citation

Díaz, J. I., Gómez-Castro, D., Shaposhnikova, T. A., & Zubova, M. N. (2017). Change of homogenized absorption term in diffusion processes with reaction on the boundary of periodically distributed asymmetric particles of critical size. <i>Electronic Journal of Differential Equations, 2017</i>(178), pp. 1-25.

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