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

作者：Musolino Paolo^*

来源：Mathematical Methods in the Applied Sciences, 2012, 35(3): 334-349.

DOI：10.1002/mma.1575

• Boundary value problems for second-order elliptic equations
• integral representations
• integral operators
• integral equations methods
• singularly perturbed domain
• Laplace operator
• periodically perforated domain
• real analytic continuation in Banach space

Let O be a sufficiently regular bounded connected open subset of R-n such that 0 epsilon Omega and that Rn\clO is connected. Then we take q11,......,q(nn) epsilon]0,+infinity[and p epsilon Q Pi(n)(j=1)]0,qjj[center dot If e is a small positive number, then we define the periodically perforated domain S[Omega(epsilon)](-) R-n\U(z epsilon)Z(n)cl (p+epsilon Omega+Sigma(n)(j=1) (qjjzj)ej, where {e(1),.....e(n)} is the canonical basis of Rn. For e small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set 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 e and of the Dirichlet datum on p+epsilon partial derivative Omega, around a degenerate pair with epsilon=0.

• 出版日期2012-2