Homogenization of an Elliptic Equation in a Domain with Oscillating Boundary with Non-homogeneous Non-linear Boundary Conditions

Abstract

While considering boundary value problems with oscillating coefficients or in oscillating domains, it is important to associate an asymptotic model which accounts for the average behaviour. This model permits to obtain the average behaviour without costly numerical computations implied by the fine scale of oscillations in the original model. The asymptotic analysis of boundary value problems in oscillating domains has been extensively studied and involves some key issues such as: finding uniformly bounded extension operators for function spaces on oscillating domains, the choice of suitable sequences of test functions for passing to the limit in the variational formulation of the model equations etc. In this article, we study a boundary value problem for the Laplacian in a domain, a part of whose boundary is highly oscillating (periodically), involving non-homogeneous non-linear Neumann or Robin boundary condition on the periodically oscillating boundary. The non-homogeneous Neumann condition or the Robin boundary condition on the oscillating boundary adds a further difficulty to the limit analysis since it involves taking the limits of surface integrals where the surface changes with respect to the parameter. Previously, some model problems have been studied successfully in Gaudiello (Ricerche Mat 43(2):239–292, 1994) and in Mel’nyk (Math Methods Appl Sci 31(9):1005–1027, 2008) by converting the surface term into a volume term using auxiliary boundary value problems. Some problems of this nature have also been studied using an extension of the notion of two-scale convergence (Allaire et al. in Proceedings of the international conference on mathematical modelling of flow through porous media, Singapore, 15–25, 1996, Neuss-Radu in C R Acad Sci Paris Sr I Math 322:899–904, 1996). In this article, we use a different approach to handle of such terms based on the unfolding operator.