TY - JOUR

T1 - The Dirichlet problem in a planar domain with two moderately close holes

AU - Dalla Riva, Matteo

AU - Musolino, Paolo

PY - 2017/9/5

Y1 - 2017/9/5

N2 - We investigate a Dirichlet problem for the Laplace equation in a domain of R2 with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance |ϵ1| one from the other and each one of size |ϵ1ϵ2|. In such a domain, we introduce a Dirichlet problem and we denote by uϵ1,ϵ2 its solution. We show that the dependence of uϵ1,ϵ2 upon (ϵ1,ϵ2) can be described in terms of real analytic maps of the pair (ϵ1,ϵ2) defined in an open neighbourhood of (0,0) and of logarithmic functions of ϵ1 and ϵ2. Then we study the asymptotic behaviour of uϵ1,ϵ2 as ϵ1 and ϵ2 tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between ϵ1 and ϵ2.

AB - We investigate a Dirichlet problem for the Laplace equation in a domain of R2 with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance |ϵ1| one from the other and each one of size |ϵ1ϵ2|. In such a domain, we introduce a Dirichlet problem and we denote by uϵ1,ϵ2 its solution. We show that the dependence of uϵ1,ϵ2 upon (ϵ1,ϵ2) can be described in terms of real analytic maps of the pair (ϵ1,ϵ2) defined in an open neighbourhood of (0,0) and of logarithmic functions of ϵ1 and ϵ2. Then we study the asymptotic behaviour of uϵ1,ϵ2 as ϵ1 and ϵ2 tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between ϵ1 and ϵ2.

KW - Dirichlet problem

KW - singularly perturbed perforated planar domain

KW - moderately close holes

KW - laplace operator

KW - real analytic continuation in Banach space

KW - asymptotic expansion

UR - http://hdl.handle.net/2160/45182

U2 - 10.1016/j.jde.2017.04.006

DO - 10.1016/j.jde.2017.04.006

M3 - Article

SN - 0022-0396

VL - 263

SP - 2567

EP - 2605

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 5

ER -