Periodic Reiterated Homogenization of Monotone Operator

Nicolas Meunier MAP5, Université Paris Descartes and IUFM Paris

                 email: meunier@ann.jussieu.fr

Abstract: We study periodic reiterated homogenization for equations of the form −div(a_{ϵ}(x, Du_{ϵ})) = f, where a_{ε} is a Carathéodory function. Under appropriate growth and monotonicity assumptions and if the sequence of reiterated unfolding converges almost everywhere to a Carathéodory type function, the sequence of solutions converges to the solution of a limit variational problem. In particular this contains the case a_{ϵ}(x, ξ) = a(x, \frac{x}{ϵ}, \frac{\{ \frac{x}{ϵ} \}_{Y}}{δ(ϵ)} , ξ), where a is periodic in the second and third arguments, and continuous in each argument.

We also study the homogenization in the monotone multivalued case for equations of the form −div d_{ε} = f, with (∇u_{ε}(x), d_{ε}(x)) ∈ A_{ε}(x), where A_{ε} is a function whose values are maximal monotone.