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

Abstract

Let $\Omega$ be a sufficiently regular bounded open connected subset of ${\mathbb{R}}^n$ such that $0\in \Omega$ and that ${\mathbb{R}}^n\setminus \mathrm{cl}\Omega$ is connected. Then we take $q_{11},...,q_{nn}\in ]0,+\infty [$ and $p\in Q\equiv {\prod }_{j=1}^n]0,q_{jj}[$. If $\epsilon$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[{\Omega }_{\epsilon }{]}^{-}\equiv {\mathbb{R}}^n\setminus {\cup }_{z\in {\mathbb{Z}}^n}\mathrm{cl}(p+\epsilon \Omega +{\sum }_{j=1}^n(q_{jj}{z}_j)e_j)$, where $\{e_1,...,e_n\}$ is the canonical basis of ${\mathbb{R}}^n$. For $\epsilon$ small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set $\mathbb{S}[{\Omega }_{\epsilon }{]}^{-}$. Namely, we consider a Dirichlet condition on the boundary of the set $p+\epsilon \Omega$, together with a periodicity condition. Then we show real analytic continuation properties of the solution and of the corresponding energy integral as functionals of the pair of $\epsilon$ and of the Dirichlet datum on $p+\epsilon \partial \Omega$, around a degenerate pair with $\epsilon =0$.

References in corpus (1)

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

Cited by in corpus (3)

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