Existence results for a nonlinear nonautonomus transmission problem via domain perturbation

TL;DR

This paper investigates the existence and analytic dependence of solutions to a nonlinear nonautonomous transmission problem for the Laplace equation, focusing on how solutions change with domain perturbations involving a perforated domain and an inclusion.

Contribution

It provides new results on the existence and analytic dependence of solutions under domain perturbations for a nonlinear transmission problem involving the Laplace equation.

Findings

Established existence of solutions for fixed inclusion.

Demonstrated analytic dependence of solutions on domain perturbation.

Analyzed the effect of shape changes on boundary value problems.

Abstract

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$.