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}/clΩ is connected. Then we choose a point w∊]0, 1 [^{n}. If e is a small positive real number, then we define the periodically perforated domain T(∊)≡R^{n}/∪_{z∊Zn}cl(w+∊Ω+z). For each small positive ∊, we introduce a particular Dirichlet problem for the Laplace operator in the set T(∊). More precisely, we consider a Dirichlet condition on the boundary of the set w+∊Ω, and we denote the unique periodic solution of this problem by u[∊]. Then we show that (suitable restrictions of) u[∊] can be continued real analytically in the parameter ∊ around ∊ = 0.
 Publication:

Icnaam 2010: International Conference of Numerical Analysis and Applied Mathematics 2010
 Pub Date:
 September 2010
 DOI:
 10.1063/1.3498645
 arXiv:
 arXiv:1307.3023
 Bibcode:
 2010AIPC.1281..928M
 Keywords:

 Laplace equations;
 boundaryvalue problems;
 functional analysis;
 integral equations;
 02.30.Vv;
 02.60.Lj;
 02.30.Sa;
 02.30.Rz;
 Operational calculus;
 Ordinary and partial differential equations;
 boundary value problems;
 Functional analysis;
 Integral equations;
 Mathematics  Analysis of PDEs
 EPrint:
 Numerical analysis and applied mathematics. ICNAAM 2010, Rhodes, Greece, 1925 September 2010, AIP Conference Proceedings vol. 1281, pages 928931. American Institute of Physics, Melville, NY, 2010