### Homogenization of a quasi-linear problem with quadratic growth in perforated domains : an example

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In this paper, a ${W}^{-1,{N}^{\text{'}}}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on ${\mathbb{R}}^{N}$, or on a regular bounded open set of ${\mathbb{R}}^{N}$. The proof is based partially on the Strauss inequality [Strauss, 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [ 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and to prove an...

In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for scalar equations does not hold for vector-valued systems, we present an alternative approach based on an approximation result by Lipschitz functions due to...

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.

In this paper, a ${W}^{-1,{N}^{\text{'}}}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on ${\mathbb{R}}^{N}$, or on a regular bounded open set of ${\mathbb{R}}^{N}$. The proof is based partially on the Strauss inequality [Strauss, (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [ (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of...

For a fixed bounded open set $\Omega \subset {\mathbb{R}}^{N}$, 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.

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