From Steklov to Neumann via homogenisation.

In this paper we study the relationship between the Steklov and Neumann eigenvalue problems on a Euclidean domain. This is done through a non-standard homogenisation limit of the Steklov problem converging to a family of eigenvalue problems with dynamical boundary conditions. In that case, the spectral parameter appears in both the interior and the boundary of the domain. It is then proved that this intermediary problem interpolates between the Steklov and the Neumann eigenvalues of the domain. This is used to create a correspondance between isoperimetric type bounds for the Steklov and Neumann eigenvalues, which sheds a new light on the question of optimal upper bounds for Steklov eigenvalues under a perimeter constraint. The proofs are based on a modification of the energy method and require careful quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.