Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem

TL;DR

This paper investigates the local uniqueness of solutions to a nonlinear transmission problem involving the Laplace equation with a small inclusion, highlighting conditions under which solutions are unique for small perturbations.

Contribution

It establishes local uniqueness results for solutions to a nonlinear, nonautonomous transmission problem with a small inclusion, extending understanding of singular perturbations.

Findings

Solutions exist for sufficiently small inclusion size

Local uniqueness holds under certain conditions

Analysis extends to nonautonomous nonlinear transmission problems

Abstract

We consider the Laplace equation in a domain of $\mathbb{R}^n$, $n\ge 3$, with a small inclusion of size $\epsilon$. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For $\epsilon$ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.