  We study a new link between the Steklov and Neumann eigenvalues of domains in
Euclidean space. This is obtained through an homogenisation limit of the
Steklov problem on a periodically perforated domain, converging to a family of
eigenvalue problems with dynamical boundary conditions. For this problem, the
spectral parameter appears both in the interior of the domain and on its
boundary. This intermediary problem interpolates between Steklov and Neumann
eigenvalues of the domain. As a corollary, we recover some isoperimetric type
bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov
eigenvalues. The interpolation also leads to the construction of planar domains
with first perimeter-normalized Stekov eigenvalue that is larger than any
previously known example. The proofs are based on a modification of the energy
method. It requires quantitative estimates for norms of harmonic functions. An
intermediate step in the proof provides a homogenisation result for a
transmission problem.
