Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole

Let A be the annular domain obtained by removing from a bounded open domain Io of Rn a small cavity of size-> 0. Then we assume that for some natural index l, λl [Io] > 0 is a simple Neumann eigenvalue of δ in Io, and we show that there exists a real valued real analytic function ̂λl(, ) defined in an open neighborhood of (0, 0) in R2 such that the lth Neumann eigenvalue λl [A] of δ in A equals ̂λl(n log ) and such that ̂λl(0, 0) = λl [Io]. Here n = 1 if n is even and n = 0 if n is odd. Thus in particular, we show that if n is even λl [A] can be expanded into a convergent double series of powers of-and-log-and that if n is odd λl [A] can be expanded into a convergent series of powers of. Then related statements have been proved for corresponding eigenfunctions. © Revista Matemática Complutense 2011.