We consider a heat transmission problem for a composite material which fills the $n$-dimensional Euclidean space. The composite has a periodic structure and consists of two materials. In each periodicity cell one material occupies a cavity of size $epsilon$, and the second material fills the remaining part of the cell. We assume that the thermal conductivities of the materials depend nonlinearly upon the temperature. We show that for $epsilon$ small enough the problem has a solution, extiti.e., a pair of functions which determine the temperature distribution in the two materials. Then we analyze the behavior of such a solution as $epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. In particular we prove that if $ngeq 3$, the temperature can be expanded into a convergent series expansion of powers of $epsilon$ and that if $n=2$ the temperature can be expanded into a convergent double series expansion of powers of $epsilon$ and $epsilonlogepsilon$.
|Titolo:||A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach|
|Data di pubblicazione:||2014|
|Appare nelle tipologie:||2.1 Articolo su rivista |
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