Sharp convergence rate of eigenvalues in a domain with a shrinking tube

Abstract

In this paper we consider a class of singularly perturbed domains, obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero. We use an Almgren-type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues, via Courant-Fischer Min-Max characterization and blow-up analysis for scaled eigenfunctions.

A note on the supersolution method for Hardy’s inequality

“…Namely, we will see that adding a suitable (finite) number of thin tubes on the flat part of ∂H will decrease the quantity ρ(H). In this part, we will greatly rely on the technique recently studied (in greater generality) in [1] by L. Abatangelo and the third author (see also [17]). We point out that, in order to apply the results of [1], we will need to know that λ 1 (H) is attained, i.e.…”

Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region

“…From [1] we recall the following result, regarding the maximum of quadratic forms with coefficients depending on a parameter (see also [13]).…”