On elliptic operators with Steklov condition perturbed by Dirichlet condition on a small part of boundary

We consider a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition. The problem involves a singular perturbation, which is the Dirichlet condition imposed on a small piece of the boundary. We rewrite such problem to a resolvent equation for a self-adjoint operator in a fractional Sobolev space on the boundary of the domain. We prove the norm convergence of this operator to a limiting one associated with an unperturbed problem involving no Dirichlet condition. We also establish an order sharp estimate for the convergence rate. The established convergence implies the convergence of the spectra and spectral projectors. In the second part of the work we study perturbed eigenvalues converging to limiting simple discrete ones. We construct two-terms asymptotic expansions for such eigenvalues and for the associated eigenfunctions.

Automated summary

The study focuses on a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition, involving singular perturbation and Dirichlet condition. It demonstrates the norm convergence of an operator related to the unperturbed problem, establishes a sharp estimate for the convergence rate, and shows the convergence of spectra and spectral projectors. Furthermore, the research examines perturbed eigenvalues converging to simple discrete limiting ones, constructing two-terms asymptotic expansions for these eigenvalues and associated eigenfunctions.