# Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains

created on 22 May 2002
modified by dalmaso on 22 Dec 2004

[BibTeX]

Published Paper

Inserted: 22 may 2002
Last Updated: 22 dec 2004

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
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.