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 -