Abstract
Abstract The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which {1<p<n} , {n\geq 3} . In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.
Highlights
The aim of this paper is to consider the asymptotic behavior of boundary value problems in ndimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size
In contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which includes the case of microscopic Dirichlet boundary conditions
A well-known effect in homogenization theory is the appearance of some changes in the structural modelling of the homogenized problem for suitable critical size of the elements configuring the “micro-structured” material
Summary
A well-known effect in homogenization theory is the appearance of some changes in the structural modelling of the homogenized problem for suitable critical size of the elements configuring the “micro-structured” material. Thanks to this generality on the maximal monotone graph σ, our treatment includes the case of microscopic Dirichlet boundary conditions In this way we unify the treatment of apparently different formulations, which before were considered separately. The generality assumed on the maximal monotone graph σ of R2 allows to treat, in a unified way, different cases as the case of Dirichlet boundary conditions, which corresponds to the choice of σ given by. Some previous results by the authors [11], formulated there for some not necessarily Lipschitz functions σ and p ∈ [2, n), will be here extended to the case a general maximal monotone graph σ (which includes the case of Dirichlet boundary conditions) and p ∈ (1, n).
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