Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 53, pp. 1-29.

Title: Existence of solutions for a singularly perturbed nonlinear non-autonomous
       transmission problem

Author: Riccardo Molinarolo (Aberystwyth Univ., Aberystwyth, Ceredigion SY23, UK)

Abstract:
 In this article we analyze a boundary value problem for the Laplace equation
 with a nonlinear non-autonomous transmission conditions on the boundary of
 a small inclusion of size $\epsilon$. We show that the problem has solutions
 for $\epsilon$ small enough and we investigate the dependence of a specific
 family of solutions upon $\epsilon$. By adopting a functional analytic
 approach we prove that the map which takes $\epsilon$ to
 (suitable restrictions of) the corresponding solution  can be represented
 in terms of real analytic functions.

Submitted May 8, 2018. Published April 22, 2019.
Math Subject Classifications: 35J25, 31B10, 45A05, 35B25, 35C20.
Key Words: Nonlinear non-autonomous transmission problem;
           singularly perturbed perforated domain; small inclusion; Laplace operator;
           real analytic continuation in Banach space; asymptotic behaviour.