Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach
S. V. Gryshchuk, Massimo Lanza de Cristoforis · Mathematical Methods in the Applied Sciences · 2013
Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small-cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all ε > 0 small enough, and such that . Here denotes the outward unit normal of , 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.