A singularly perturbed Dirichlet problem for the Poisson equation 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 $(q_{11},\dots ,q_{nn})\in ]0,+\infty {[}^n$ 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 $\mathbb{S}[{\Omega }_{p,\epsilon }{]}^{-}\equiv {\mathbb{R}}^n\setminus {\cup }_{z\in {\mathbb{Z}}^n}\mathrm{cl}(p+\epsilon \Omega +{\sum }_{j=1}^n(q_{jj}{z}_j)e_j)$, where $\{e_1,\dots ,e_n\}$ is the canonical basis of ${\mathbb{R}}^n$. For $\epsilon$ small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set $\mathbb{S}[{\Omega }_{p,\epsilon }{]}^{-}$. 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 as a function of $\epsilon$, of the Dirichlet datum on $p+\epsilon \partial \Omega$, and of the Poisson datum, around a degenerate triple with $\epsilon =0$.

arXiv admin note: substantial text overlap with arXiv:1306.6674

References in corpus (1)

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

Cited by in corpus (1)

• A singularly perturbed nonlinear traction problem in a periodically perforated domain. A functional analytic approach