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In this paper we study a control problem for elliptic nonlinear monotone problems with Dirichlet boundary conditions where the control variables are the coefficients of the equation and the open set where the partial differential problem is studied.


For a fixed bounded open set $\Omega \subset {ℝ}^N$, a sequence of open sets ${\Omega }_n\subset \Omega$ and a sequence of sets ${\Gamma }_n\subset \partial \Omega \cap \partial {\Omega }_n$, we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on ${\Omega }_n$, satisfying Neumann boundary conditions on ${\Gamma }_n$ and Dirichlet boundary conditions on $\partial {\Omega }_n\setminus {\Gamma }_n$. We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on ${\Omega }_n$ and ${\Gamma }_n$ locally.