Periodic Reiterated Homogenization of Monotone Operator

Nicolas Meunier
MAP5, Université Paris Descartes and IUFM Paris

Abstract: We study periodic reiterated homogenization for equations of the form $-\textrm{div}(a_\epsilon (x,Du_\epsilon ))=f$, where $a_\varepsilon$ 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 $a_\epsilon (x,\xi)=a(x,\frac{x}{\epsilon
},\frac{\{\frac{x}{\epsilon}\}_Y}{ \delta (\epsilon )},\xi)$, 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 $-\mathrm{div}\;d_\varepsilon=f $, with $\bigl(\nabla
u_\varepsilon(x),d_\varepsilon(x)\bigr) \in A_\varepsilon(x)$, where $A_\varepsilon $ is a function whose values are maximal monotone.