*Published Paper*

**Inserted:** 22 may 2002

**Last Updated:** 22 dec 2004

**Journal:** Ann. Inst. H. PoincarĂ© Anal. Non LinĂ©aire

**Volume:** 21

**Pages:** 445-486

**Year:** 2004

**Abstract:**

We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in ${\cal M}_0^+(\Omega)$) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices $H$-converge to a matrix $A^0$, we prove that there exist a subsequence and a measure $\mu^0$ in ${\cal M}_0^+(\Omega)$ such that the limit problem is the relaxed Dirichlet problem corresponding to $A^0$ and $\mu^0$. We also prove a corrector result which provides an explicit approximation of the solutions in the $H^{1}$-norm, and which is obtained by multiplying the corrector for the $H$-converging matrices by some special test function which depends both on the varying matrices and on the varying domains.

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