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.

We investigate a Dirichlet problem for the Laplace equation in a domain of R-2 with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance |epsilon(1)| one from the other and each one of size |epsilon(1)epsilon(2)|. In such a domain, we introduce a Dirichlet problem and we denote by u(epsilon 1),(epsilon 2) its solution. We show that the dependence of u epsilon(1),epsilon(2) upon (epsilon(1), epsilon(2)) can be described in terms of real analytic maps of the pair (epsilon(1), epsilon(2)) defined in an open neighbourhood of (0, 0) and of logarithmic functions of epsilon(1) and epsilon(2). Then we study the asymptotic behaviour of u(epsilon 1),(epsilon 2) as epsilon(1) and epsilon(2) 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 epsilon(1) and epsilon(2). (C) 2017 Published by Elsevier Inc.

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

Musolino P.
2017

Abstract

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.
File in questo prodotto:
File Dimensione Formato  
170307_dirclose.pdf

non disponibili

Descrizione: preprint
Tipologia: Documento in Pre-print
Licenza: Accesso chiuso-personale
Dimensione 381.72 kB
Formato Adobe PDF
381.72 kB Adobe PDF   Visualizza/Apri

I documenti in ARCA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10278/3723524
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 2
social impact