Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach

Abstract

We consider a bounded open subset 𝕀 o of ℝ n with 0 ∈ 𝕀 o , and a function f o of ∂𝕀 o to ℝ. Under reasonable assumptions, the Dirichlet problem Δu = 0 in 𝕀 o , u = f o on ∂𝕀 o , has one and only one solution ũ o . Then we consider another bounded open subset 𝕀 i of ℝ n with 0 ∈ 𝕀 i , and an increasing diffeomorphism F of ℝ onto itself, and a constant γ ∈]0, +∞[, and a function g of ∂𝕀 i to ℝ, and we consider the non-linear transmission boundary value problem for ε > 0 small, where νε𝕀 i is the outward unit normal to ε∂𝕀 i . Under suitable conditions on the data, we show that for sufficiently small, such a boundary value problem admits locally around (F (−1)(ũ o (0)), ũ o ) a family of solutions . Then we show that u i (ε, ε·) and (suitable restrictions of) u o (ε, ·) and u o (ε, ε·) can be continued real analytically in the parameter ε around ε = 0 for n ≥ 3, and can be represented in terms of real analytic functions of ε, log−1 ε, ε log2 ε for n = 2.