Interactions between moderately close circular inclusions: the Dirichlet-Laplace equation in the plane
Résumé
The presence of small inclusions or of a surface defect modifies the solution of the Laplace equation posed in a reference domain. If the characteristic size of the perturbation is small, then one can expect that the solution of the problem posed on the perturbed geometry is close to the solution of the reference shape. Asymptotic expansion with respect to that small parameter -the characteristic size of the perturbation- can then be performed. We consider in the present work the case of two defects with Dirichlet boundary conditions in a bidimensional domain. For the simplicity of the presentation, we assume that the defects we are considering are disks. We build an asymptotic expansion of the solution of the Laplace problem in perturbed domains. We will consider two unstudied cases: In the first case, we are considering two small holes around two fixed points (the distance between both is hence fixed). For the cases of Neumann boundary condition or of Dirichlet boundary conditions in dimension at least three, this cases can be treated by separating each hole through cut-off functions and hence reducing it to the single inclusion case. Here, the presence of the logarithmic term prohibits this approach and the interaction between the holes has to be studied. In the second case, the distance between the centers collapses to 0 slower than the size of the inclusions. The interaction between the two holes are then stronger and we will prove that the leading order of the asymptotic expansion is then modified.
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