Let \$Omega\$ be a sufficiently regular bounded open connected subset of \$mathbbR^n\$ such that \$0 in Omega\$ and that \$mathbbR^n setminus mathrmclOmega\$ is connected. Then we take \$(q_11,dots, q_nn)in ]0,+infty[^n\$ and \$p in Qequiv prod_j=1^n]0,q_jj[\$. If \$epsilon\$ is a small positive number, then we define the periodically perforated domain \$mathbbS[Omega_p,epsilon]^- equiv mathbbR^nsetminus cup_z in mathbbZ^nmathrmcligl(p+epsilon Omega +sum_j=1^n (q_jjz_j)e_jigr)\$, where \$e_1,dots,e_n\$ is the canonical basis of \$mathbbR^n\$. For \$epsilon\$ small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set \$mathbbS[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\$.

### 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 \$mathbbR^n\$ such that \$0 in Omega\$ and that \$mathbbR^n setminus mathrmclOmega\$ is connected. Then we take \$(q_11,dots, q_nn)in ]0,+infty[^n\$ and \$p in Qequiv prod_j=1^n]0,q_jj[\$. If \$epsilon\$ is a small positive number, then we define the periodically perforated domain \$mathbbS[Omega_p,epsilon]^- equiv mathbbR^nsetminus cup_z in mathbbZ^nmathrmcligl(p+epsilon Omega +sum_j=1^n (q_jjz_j)e_jigr)\$, where \$e_1,dots,e_n\$ is the canonical basis of \$mathbbR^n\$. For \$epsilon\$ small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set \$mathbbS[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\$.
##### Scheda breve Scheda completa Scheda completa (DC)
2013
Advances in Harmonic Analysis and Operator Theory, The Stefan Samko Anniversary Volume
File in questo prodotto:
File
dirpoiper24nov11-revision.pdf

non disponibili

Descrizione: preprint
Tipologia: Documento in Pre-print
Licenza: Accesso chiuso-personale
Dimensione 249.52 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10278/3723639`