Homogenization of a class of singular elliptic problems in two-component domains

Abstract

This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ${\mathit{\epsilon }}^{\mathit{\gamma }}$ .

We prove an homogenization result for $-1\mathrm{<}\mathit{\gamma }\mathrm{<}1$ by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 (2008) 1585–1620 and The Periodic Unfolding Method (2018) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011) 891–927).

One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence $\{{\mathit{u}}^{\mathit{\epsilon }}\}$ of the solutions, as $\mathit{\epsilon }\to 0$ . More precisely, our result shows that the gradient of ${\mathit{u}}^{\mathit{\epsilon }}$ behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate.