Existence results for a nonlinear nonautonomous transmission problem via domain perturbation

Matteo Dalla Riva,Riccardo Molinarolo,Paolo Musolino

doi:10.1017/prm.2021.60

Matteo Dalla Riva, Riccardo Molinarolo + Show 1 more

Open Access

https://doi.org/10.1017/prm.2021.60

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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 $\phi$.

Highlights

We begin by introducing the geometric framework of our problem
We introduce a ‘perturbed’ version of problem (1.3): we fix the external domain Ωo and we assume that the boundary of the internal domain is of the form φ(∂Ωi), where φ is a diffeomorphism of ∂Ωi into a subset of Rn that belongs to the class
We provide a formulation of problem (1.6) in terms of integral equations depending on the diffeomorphism φ which we rewrite into an equation of the type M [φ, μ] = 0 for an auxiliary map M : AΩ∂Ωo i × X → Y, where the variable μ is related to the densities of the integral representation of the solution

Summary

Introduction

We begin by introducing the geometric framework of our problem. We fix once and for all a natural number n ∈ N\{0, 1}. We choose to adopt the Functional Analytic Approach, which has revealed to be a powerful tool to analyse perturbed linear and nonlinear boundary value problems This method has been first applied to investigate regular and singular domain perturbation problems for elliptic equations and systems with the aim of proving real analytic dependence upon the perturbation parameter (cf Lanza de Cristoforis [16, 18, 19]). Such a transformation is achieved by exploiting classical results of potential theory, for example, integral representation of harmonic functions in terms of layer potentials. By the Implicit Function Theorem, we show the existence of a family of solutions {(uoφ, uiφ)}φ∈Q0 of (1.6) (cf. theorem 5.6) and we prove that it can be represented in terms of real analytic functions (cf. theorem 5.7)

Notation

Some preliminaries of potential theory