TY - JOUR

T1 - A quasi-linear heat transmission problem in a periodic two-phase dilute composite

T2 - A functional analytic approach

AU - De Cristoforis, Massimo Lanza

AU - Musolino, Paolo

PY - 2014/11/1

Y1 - 2014/11/1

N2 - 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.

AB - 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.

KW - Asymptotic behavior

KW - Periodic two-phase dilute composite

KW - Quasi-linear heat transmission problem

KW - Real analytic continuation in Banach space

KW - Singularly perturbed domain

UR - http://www.scopus.com/inward/record.url?scp=84904579249&partnerID=8YFLogxK

U2 - 10.3934/cpaa.2014.13.2509

DO - 10.3934/cpaa.2014.13.2509

M3 - Article

AN - SCOPUS:84904579249

SN - 1534-0392

VL - 13

SP - 2509

EP - 2542

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

IS - 6

ER -