A mixed problem for the Laplace operator in a domain with moderately close holes

ABSTRACT

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ε and we define a perforated domain Ω_ε obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ε → 0. For ε small, we denote by u_ε the solution of a mixed problem for the Laplace equation in Ω_ε. We describe what happens to u_ε as ε → 0 in terms of real analytic maps and we compute an asymptotic expansion.

KEYWORDS:

• Asymptotic expansion
• Laplace operator
• mixed problem
• moderately close holes
• real analytic continuation in Banach space
• singularly perturbed perforated domain

MATHEMATICS SUBJECT CLASSIFICATION:

• 35J25
• 31B10
• 45A05
• 35B25
• 35C20

Acknowledgments

The authors wish to thank V. Bonnaillie-Noël, M. Dambrine, and C. Lacave for several useful discussions.