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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