Interactions Between Moderately Close Inclusions for the Two-Dimensional Dirichlet–Laplacian

Journal de Mathématiques Pures et Appliquées

Dambrine³

et al. 2018

Self Cite

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair ε = (ε 1 , ε 2 ) of positive parameters, we consider a perforated domain Ω ε obtained by making a small hole of size ε 1 ε 2 in an open regular subset Ω of R n at distance ε 1 from the boundary ∂Ω. As ε 1 → 0, the perforation shrinks to a point and, at the same time, approaches the boundary. When ε → (0, 0), the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by u ε the solution of a Dirichlet problem for the Laplace equation in Ω ε . For a space dimension n ≥ 3, we show that the function mapping ε to u ε has a real analytic continuation in a neighborhood of (0, 0). By contrast, for n = 2 we consider two different regimes: ε tends to (0, 0), and ε 1 tends to 0 with ε 2 fixed. When ε → (0, 0), the solution u ε has a logarithmic behavior; when only ε 1 → 0 and ε 2 is fixed, the asymptotic behavior of the solution can be described in terms of real analytic functions of ε 1 . We also show that for n = 2, the energy integral and the total flux on the exterior boundary have different limiting values in the two regimes. We prove these results by using functional analysis methods in conjunction with certain special layer potentials.

Section: Methodology: the Functional Analytic Approachmentioning

confidence: 99%

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

Bonnaillie‐Noël

Journal de Mathématiques Pures et Appliquées

Dambrine³

et al. 2018

Self Cite

“…Moreover, by standard calculus in Banach spaces and by formula (9), we deduce the validity of (12). Clearly,…”

Section: The Periodic Simple Layer Potential and Preliminariesmentioning

confidence: 74%

“…In particular, a uniform asymptotic approximation of Green's kernel for the transmission problem for domains with small inclusions has been obtained in Maz'ya, Movchan, and Nieves [35] and Nieves [45]. Boundary value problems in domains with small holes have been also analyzed with the method of multiscale asymptotic expansions (cf., e.g., Bonnaillie-Noël, Dambrine, Tordeux, and Vial [13] and Bonnaillie-Noël, Dambrine, and Lacave [12]). Moreover, the topological-shape sensitivity analysis of the energy shape functionals for perturbations in the form of inclusions with appropriate transmission conditions can be found in Novotny and Sokołowski [46,Ch.…”

Section: Introductionmentioning

confidence: 99%

Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface

Musolino

Pukhtaievych

2019

Asymptotic Analysis

We study the asymptotic behavior of the effective thermal conductivity of a periodic two-phase dilute composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material, each of them of size proportional to a positive parameter . We assume that the normal component of the heat flux is continuous at the two-phase interface, while we impose that the temperature field displays a jump proportional to the normal heat flux. For small, we prove that the effective conductivity can be represented as a convergent power series in and we determine the coefficients in terms of the solutions of explicit systems of integral equations.

“…To do so, it clearly su ces to show that the operator M r * is invertible. If r * = 0, the invertibility follows immediately by Lemma 4.2 (3). If r * > 0, we note that…”

Section: Formulation Of Problem (8) In Terms Of Integral Equationsmentioning

confidence: 80%

“…In particular, boundary value problems in domains with moderately close holes have been deeply studied in Bonnaillie-Noël et al [4,5]; Bonnaillie-Noël and Dambrine [2]; and Bonnaillie-Noël et al [3], where the authors exploit the method of multiscale asymptotic expansions. More precisely, in [5] they carefully analyze the case when η(ǫ) = ǫ β for β ∈]0, 1[ and they provide asymptotic expansions.…”

Section: Introductionmentioning

confidence: 99%

A mixed problem for the Laplace operator in a domain with moderately close holes

Communications in Partial Differential Equations

Musolino

2016

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ? and we define a perforated domain ?? obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ????0. For ? small, we denote by u? the solution of a mixed problem for the Laplace equation in ??. We describe what happens to u? as ????0 in terms of real analytic maps and we compute an asymptotic expansion.Peer reviewe