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

Let I-o be a bounded open domain of R-n. Let nu(Io) denote the outward unit normal of partial derivative I-o. We assume that the Steklov problem Delta u = 0 in I-o and partial derivative u/partial derivative nu I-o = lambda u on partial derivative I-o has a simple eigenvalue of (lambda) over tilde. Then we consider an annular domain A(epsilon) obtained by removing from I-o a small-cavity size of epsilon > 0, and we show that under proper assumptions there exists a real valued and real analytic function (lambda) over tilde(.,.) defined in an open neighborhood of (0, 0) in R-2 and such that (lambda) over tilde (epsilon, delta 2,n is an element of log epsilon) is a simple eigen-value for the Steklov problem Delta u = 0 in A (epsilon) and partial derivative u/partial derivative nu(Lambda(epsilon)) = lambda u on partial derivative A (epsilon) for all epsilon > 0 small enough, and such that (lambda) over tilde (0, 0) = (lambda) over tilde Here nu(A(epsilon)) denotes the outward unit normal of partial derivative A(epsilon), and delta(2,2) equivalent to 1 and delta(2,n) equivalent to 0 if n >= 3. Then related statements have been proved for corresponding eigenfunctions. Copyright (C) 2013 JohnWiley & Sons, Ltd.