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Pré-Publication, Document De Travail Année : 2016

A small hole close to the boundary for the two-dimensional Laplace Dirichlet problem

Résumé

We study a Dirichlet problem in a planar domain with a small hole close to the boundary. For each pair ε = (ε1, ε2) of positive parameters, we define a perforated domain Ωε obtained by making a small perforation of size ε1ε2 in an open set. The distance of the cavity from the boundary is instead controlled by ε1. As ε1 → 0, the perforation shrinks to a point and at the same time approaches the boundary. We consider separately two cases: the case when ε tends to (0, 0) and the case when ε1 tends to 0 and ε2 is fixed. In the first case we show that the solution of a Dirichlet problem defined in Ωε displays a logarithmic behavior when ε → 0. In the second case instead, the asymptotic behavior of the solution can be described in terms of real analytic functions of ε1. We will also show that the energy integral and the total flux on the exterior boundary have a different limiting behavior in the two cases.
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Dates et versions

hal-01415983 , version 1 (13-12-2016)
hal-01415983 , version 2 (13-12-2016)
hal-01415983 , version 3 (04-07-2017)
hal-01415983 , version 4 (21-10-2017)

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Virginie Bonnaillie-Noël, Matteo Dalla Riva, Marc Dambrine, Paolo Musolino. A small hole close to the boundary for the two-dimensional Laplace Dirichlet problem. 2016. ⟨hal-01415983v2⟩
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