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Estimation de L’erreur dans des problèmes de Dirichlet ou apparat un terme étrange. (Error estimates in a Dirichlet problem where a strange term appears). (French) Zbl 0687.35007
Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 661-696 (1989).
[For the entire collection see Zbl 0671.00007.]
Let $$u^{\epsilon}$$ be the solution of the Dirichlet problem $$-\Delta u^{\epsilon}=f$$ in $$\Omega^{\epsilon}$$, $$u^{\epsilon}=0$$ on $$\partial \Omega^{\epsilon}$$, where $$\Omega^{\epsilon}$$ is obtained by removing from $$\Omega$$ small holes $$T^{\epsilon}_ i$$, periodically distributed with a period $$2\epsilon$$ in the directions of the axes, each of the holes $$T^{\epsilon}_ i$$ being obtained from a model hole by reducing it at the size $$\epsilon^{N/(N-2)}.$$ These $$u^{\epsilon}$$ converge weakly to the solution u of $$-\Delta u+\mu u=f$$ in $$\Omega$$, $$u=0$$ on $$\partial \Omega$$. The following error estimate $$\| u^{\epsilon}-(1-p^{\epsilon})u\|_{H^ 1_ 0(\Omega)}\leq Const. \epsilon)$$ is proved, where $$p^{\epsilon}$$ is the capacity potential of each small hole $$T^{\epsilon}_ i$$ in the ball of radius $$\epsilon$$ with the same center.
Reviewer: G.Morosanu

##### MSC:
 35B25 Singular perturbations in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35B15 Almost and pseudo-almost periodic solutions to PDEs
##### Keywords:
domain with many holes