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

Let IIo be a bounded open domain of ℝn. Let νIIo denote the outward unit normal of ∂IIo. We assume that the Steklov problem Δu = 0 in IIo and ∂u∂νIIo =λu on ∂IIo has a simple eigenvalue of λ̃. Then we consider an annular domain A(ε) obtained by removing from IIo a small-cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function λ̂(.,.) defined in an open neighborhood of (0,0) in ℝ2 and such that λ̂(ε,δ 2,nεlogε) is a simple eigenvalue for the Steklov problem Δu = 0 in A(ε) and ∂u/∂νA(ε)=λu on ∂A(ε) for all ε > 0 small enough, and such that λ̂(0,0)=λ̃. Here νA(ε) denotes the outward unit normal of ∂A(ε), and δ2,2 ≡ 1 and δ2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 John Wiley & Sons, Ltd.