Multiple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Article type: Research Article

Authors: Lanza de Cristoforis, Massimo^{; *}

Affiliations: Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Italy. E-mail: mldc@math.unipd.it

Correspondence: [*] Corresponding author. E-mail: mldc@math.unipd.it.

Abstract: Let α∈]0,1[. Let Ωo be a bounded open domain of Rn of class C1,α. Let νΩo denote the outward unit normal to ∂Ωo. We assume that the Steklov problem Δu=0 in Ωo, ∂u∂νΩo=λu on ∂Ωo has a multiple eigenvalue λ˜ of multiplicity r. Then we consider an annular domain Ω(ϵ) obtained by removing from Ωo a small cavity of class C1,α and size ϵ>0, and we show that under appropriate assumptions each elementary symmetric function of r eigenvalues of the Steklov problem Δu=0 in Ω(ϵ), ∂u∂νΩ(ϵ)=λu on ∂Ω(ϵ) which converge to λ˜ as ϵ tend to zero, equals real a analytic function defined in an open neighborhood of (0,0) in R2 and computed at the point (ϵ,δ2,nϵlogϵ) for ϵ>0 small enough. Here νΩ(ϵ) denotes the outward unit normal to ∂Ω(ϵ), and δ2,2≡1 and δ2,n≡0 if n⩾3. Such a result is an extension to multiple eigenvalues of a previous result obtained for simple eigenvalues in collaboration with S. Gryshchuk.

Keywords: Multiple Steklov eigenvalues and eigenfunctions, singularly perturbed domain, Laplace operator, real analytic continuation

DOI: 10.3233/ASY-201605

Journal: Asymptotic Analysis, vol. 121, no. 3-4, pp. 335-365, 2021