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Asymptotic analysis of two elliptic equations with oscillating terms. (English) Zbl 0657.35016

In a bounded, smooth open subset \(\Omega\) of \(R^ N\), is disposed an \(\epsilon\)-periodic distribution \(\cup_{i}T_{\epsilon i}\) of identical inclusions. Then, the asymptotic behaviour of the solution of each of the two problems: \[ (H_{\epsilon})\quad -\Delta u_{\epsilon}+h_{\epsilon}\chi_{\cup_{i}T_{\epsilon i}}u_{\epsilon}=f\quad in\quad \Omega,\quad u_{\epsilon}\in H^ 1_ 0(\Omega). \]
\[ (M_{\epsilon})\quad -\Delta u_{\epsilon}=f\quad in\quad \Omega \setminus \overline{\cup_{i}T_{\epsilon i}},\quad (\partial u_{\epsilon}/\partial n)+b_{\epsilon}u_{\epsilon}=0\quad on\quad \cup_{i}\partial T_{\epsilon i},\quad u_{\epsilon}\in H^ 1_ 0(\Omega), \] is studied, through epi-convergence methods. In this way, we simultaneously derive the asymptotic analysis of Neumann and Dirichlet boundary problems in open sets with holes. Critical ratios combining the size \(r_{\epsilon}\) of the inclusions and the size of the highly oscillating parameters \(h_{\epsilon}\) and \(b_{\epsilon}\) are exhibited.
Reviewer: A.Brillard

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35B20 Perturbations in context of PDEs
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