A Functional Analytic Approach for a Singularly Perturbed Dirichlet Problem for the Laplace Operator in a Periodically Perforated Domain

Abstract

We consider a sufficiently regular bounded open connected subset Ω of R n such that 0 ∈ Ω and such that R n \ cl Ω is connected. Then we choose a point w ∈]0, 1[ n . If ǫ is a small positive real number, then we define the periodically perforated domain T (ǫ) ≡ R n \ ∪ z∈Z n cl(w + ǫΩ + z). For each small positive ǫ, we introduce a particular Dirichlet problem for the Laplace operator in the set T (ǫ). More precisely, we consider a Dirichlet condition on the boundary of the set w + ǫΩ, and we denote the unique periodic solution of this problem by u[ǫ]. Then we show that (suitable restrictions of) u[ǫ] can be continued real analytically in the parameter ǫ around ǫ = 0.

Regularizing properties of space‐periodic layer heat potentials and applications to boundary value problems in periodic domains

“…In a future investigation, we plan to consider also other types of problems for the heat equation in a periodic setting, like transmission problems and problems with nonlinear boundary conditions. In addition, the results of this paper are a first necessary step in order to study singular perturbation problems for the heat equation in periodically perforated domains with the potential theoretic approach which stems from Lanza de Cristoforis and which has been extended by Lanza de Cristoforis and collaborators to periodic elliptic problems (see, eg, Musolino and Lanza de Cristoforis & Musolino).…”

“…For example, Shcherbina has introduced periodic layer potentials to solve boundary value problems for the Laplace equation. Ammari et al have used periodic layer potentials for deriving the effective properties of isotropic composite materials, while the anisotropic case is considered in Ammari et al Potential theoretic methods to study singular perturbation problems for the Laplace equation in periodically perforated domains have been used for example in Musolino and Lanza de Cristoforis and Musolino . These methods have been extended also to treat different partial differential equations; see, eg, Ammari et al and Dalla Riva and Musolino for the Lamé equations and Lanza de Cristoforis and Musolino for a quasi‐linear differential equation.…”

A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach

“…The author wishes to thank Professor M. Lanza de Cristoforis for his constant help during the preparation of this paper. The results presented here have been announced in [41]. The author acknowledges the support of the research project 'Un approccio funzionale analitico per problemi di omogeneizzazione in domini a perforazione periodica' of the University of Padova, Italy.…”

Shape analysis of the longitudinal flow along a periodic array of cylinders

“…when l is in ]0, +∞[ and φ is in a suitable class of diffeomorphisms. In order to achieve this objective, we exploit some of the results of [29], where the behavior of a (singularly) perturbed Dirichlet problem for the Laplace equation has been studied by means of periodic potentials. As we shall see, we will reduce the analysis of the solution u # [l, φ] of the Dirichlet problem (10) to that of a related integral equation.…”