Asymptotic analysis of harmonic functions in singular domains with inhomogenous Robin boundary conditions

TL;DR

This paper develops an asymptotic analysis for harmonic functions in domains with small inclusions under inhomogeneous Robin boundary conditions, extending previous theories to new boundary conditions relevant in liquid crystal models.

Contribution

It derives an asymptotic approximation for harmonic functions with Robin boundary conditions in domains with small inclusions and proves uniform bounds on the approximation error.

Findings

Derived an explicit asymptotic approximation for the harmonic function.

Proved the uniform boundedness of the approximation error.

Established the rate of convergence as a function of inclusion size and boundary parameter.

Abstract

In 1991, Vladimir Maz'ya, Serguei Nazarov and Boris Plamenevskij developed the theory of compound asymptotics for elliptic boundary value problems in singularly perturbed domains. They considered a harmonic function whose domain contains a small inclusion. We applied this technique in the analysis of a Nematic liquid crystal with a small colloidal inclusion. However, we realised that the Maz'ya, Nazarov and Plamenevskij did not consider the asymptotic analysis of Robin boundary conditions, which corresponded to weak anchoring in the context of liquid crystals. In this piece we shall derive an asymptotic approximation to a harmonic function, in a domain with a small circular inclusion of radius $\epsilon>0$, with inhomogenous Robin boundary conditions and a corresponding parameter $\kappa>0$. We shall then prove that the difference between the exact solution and the approximation is uniformly bounded and derive the rate as a function of $\epsilon$ and $\kappa$.