Singular Perturbation of Simple Steklov Eigenvalues

Let I-o be a bounded open domain of R-n. Let nu(Io) denote the outward unit normal to partial derivative I-o. We assume that the Steklov problem Delta u = 0 in I-o, partial derivative u partial derivative(nu Io) = lambda u on partial derivative I-o has a simple eigenvalue (lambda) over tilde. Then we consider an annular domain A(epsilon) obtained by removing from I-o a small cavity of size epsilon > 0, and we show that under proper assumptions there exists a real valued real analytic function (lambda) over cap(.,.) defined in an open neighborhood of (0,0) in R-2 and such that (lambda) over cap(epsilon, delta(2,n)epsilon log epsilon) is a simple eigenvalue for the Steklov problem Delta u = 0 in A(epsilon), partial derivative u/partial derivative nu(A(epsilon)) = lambda u on partial derivative A(epsilon) for all epsilon > 0 small enough, and such that (lambda) over cap (0,0) = (lambda) over tilde. Here nu(A(epsilon)) denotes the outward unit normal to 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.