A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach

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 $epsilonlogepsilon$.

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 ε, 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 ε small enough the problem has a solution, i.e., a pair of functions which determine the temperature distribution in the two materials. Then we analyze the behavior of such a solution as ε approaches 0 by an approach which is alternative to those of asymptotic analysis. In particular we prove that if n ≥ 3, the temperature can be expanded into a convergent series expansion of powers of ε and that if n = 2 the temperature can be expanded into a convergent double series expansion of powers of ε and ε log ε.