We consider a sequence of Dirichlet problems in varying domains(or, more generally, of relaxed Dirichlet problems involving measures in ${\mathcal M}_{0}^{+}(\Omega)$) 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 ${\mathcal M}_{0}^{+}(\Omega)$ 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.
Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains
Dal Maso, Gianni;
2004-01-01
Abstract
We consider a sequence of Dirichlet problems in varying domains(or, more generally, of relaxed Dirichlet problems involving measures in ${\mathcal M}_{0}^{+}(\Omega)$) 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 ${\mathcal M}_{0}^{+}(\Omega)$ 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.File | Dimensione | Formato | |
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