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_nnin ]0,+infty[\$ 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_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 Laplace operator in the set \$mathbbS[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\$.

Let Ω be a sufficiently regular bounded connected open subset of R n such that 0 ∈ Ω and that R n\clΩ is connected. Then we take q 11, ⋯ ,q nn ∈ ]0,+ ∞ [and p∈Q≡∏ j=1n]0,q jj[. If ε is a small positive number, then we define the periodically perforated domain S[Ω ε]-≡R n\ ∪ z∈Zn/cl(p+εΩ+∑ j=1n(q jjz j)e j, where {e 1, ⋯ ,e n} is the canonical basis of R n. For ε small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set S[Ωε]-. Namely, we consider a Dirichlet condition on the boundary of the set p + εΩ, 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 ε and of the Dirichlet datum on p + ε∂Ω, around a degenerate pair with ε = 0. © 2011 John Wiley & Sons, Ltd.

### 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 \$mathbbR^n\$ such that \$0 in Omega\$ and that \$mathbbR^n setminus mathrmclOmega\$ is connected. Then we take \$q_11,dots, q_nnin ]0,+infty[\$ 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_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 Laplace operator in the set \$mathbbS[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\$.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/10278/3723628`