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 ν_{ε𝕀 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.

Keywords:

• non-linear transmission problem
• singularly perturbed domain
• Laplace operator
• real analytic continuation in Banach space

AMS Subject Classifications::

• 35J25
• 31B10
• 45F15
• 47H30

Acknowledgement

The author is indebted to V.V. Mityushev and S.V. Rogosin for several discussions on transmission problems in the frame of the applications, and for pointing out to the author problem (2).