A Local Uniqueness Result for a Quasi-linear Heat Transmission Problem in a Periodic Two-phase Dilute Composite

We consider a quasi-linear 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. For $epsilon$ small enough the problem is known to have a solution, extiti.e., a pair of functions which determine the temperature distribution in the two materials. Then we prove a limiting property and a local uniqueness result for families of solutions which converge as $epsilon$ tends to $0$.