Electron. J. Differential Equations,
Vol. 2017 (2017), No. 178, pp. 125.
Change of homogenized absorption term in diffusion processes with
reaction on the boundary of periodically distributed
asymmetric particles of critical size
Jesus Ildefonso Diaz, David GomezCastro,
Tatiana A. Shaposhnikova, Maria N. Zubova
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)
,
in the ndimensional space, are diffeomorphic
to a ball (of diameter
,
for some
.
To define the corresponding "new strange term"
we introduce a oneparametric family of auxiliary external problems
associated to canonical cellular problem associated to the prescribed
asymmetric geometry
and the nonlinear microscopic boundary reaction
(which is assumed to be merely a Holder continuous function).
We construct the limit homogenized problem and prove that it is a
wellposed global problem, showing also the rigorous convergence of solutions,
as
,
in suitable functional spaces.
This improves many previous papers in the literature dealing with
symmetric particles of critical size.
Submitted June 12, 2017. Published July 13, 2017.
Math Subject Classifications: 35B25, 35B40, 35J05, 35J20
Key Words: Homogenization; diffusion processes; periodic asymmetric particles;
microscopic nonlinear boundary reaction; critical sizes.
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Jesús Ildefonso Díaz
Instituto de Matematica Interdisciplinar
Universidad Complutense de Madrid
28040 Madrid, Spain
email: ildefonso.diaz@mat.ucm.es


David G&ocute;mezCastro
Instituto de Matematica Interdisciplinar
Universidad Complutense de Madrid
28040 Madrid, Spain
email: dgcastro@ucm.es


Tatiana A. Shaposhnikova
Faculty of Mechanics and Mathematics
Moscow State University
Moscow, Russia
email: shaposh.tan@mail.ru


Maria N. Zubova
Faculty of Mechanics and Mathematics
Moscow State University
Moscow, Russia
email: zubovnv@mail.ru

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