On a multiscale analysis of an inverse problem of nonlinear transfer law identification in periodic microstructure

This paper is devoted to the multiscale analysis of a homogenization inverse problem of the heat exchange law identification, which is governed by parabolic equations with nonlinear transmission conditions in a periodic heterogeneous medium. The aim of this work is to transform this inverse problem with nonlinear transmission conditions into a new one governed by a less complex nonlinear parabolic equation, while preserving the same form and physical properties of the heat exchange law that it will be identified, based on periodic homogenization theory. For this, we reformulate first the encountered homogenization inverse problem to an optimal control one. Then, we study the well-posedness of the state problem using the Leray–Schauder topological degrees and we also check the existence of the solution for the obtained optimal control problem. Finally, using the periodic homogenization theory and priori estimates, with justified choise of test functions, we reduce our inverse problem to a less complex one in a homogeneous medium.