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

TL;DR

This paper investigates the existence and behavior of solutions to a nonlinear boundary value problem involving a small inclusion, demonstrating solutions exist for small perturbations and depend analytically on the perturbation parameter.

Contribution

It establishes the existence of solutions for a singularly perturbed nonlinear transmission problem and shows the solution map is real analytic in the perturbation parameter.

Findings

Solutions exist for sufficiently small inclusion size

The solution map is real analytic with respect to the perturbation parameter

Dependence of solutions on the inclusion size is analytically characterized

Abstract

In this paper we analyse 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.