Compensated compactness for nonlinear homogenization and reduction of dimension

Abstract.

We study the limit behaviour of some nonlinear monotone equations, such as: $-div(A^\epsilon \varphi (B^\epsilon \nabla U^\epsilon)) = F^\epsilon$, in a domain $\Omega^\epsilon$ which is thin in some directions (e.g. $\Omega^\epsilon$ is a plate or a thin cylinder). After rescaling to a fixed domain $\Omega$, the above equation is transformed into: $-div^\epsilon(a^\epsilon \varphi (b^\epsilon \nabla^\epsilon u^\epsilon)) = f^\epsilon$, with convenient operators $div^\epsilon$ and $\nabla^\epsilon$. Assuming that $a^\epsilon$ and the inverse of $b^\epsilon$ have particular forms and satisfy suitable compensated compactness assumptions, we prove a closure result, that is we prove that the limit problem has the same form. This applies in particular to the limit behaviour of nonlinear monotone equations in laminated plates.