Abstract
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.
Highlights
Let ⊂ Rd be a bounded and connected domain with Lipschitz boundary ∂
Let us start by painting with a broad brush the relationships between the Neumann and Steklov eigenvalue problems; they exhibit many similar features, and it is not a surprise that they do so
The regime that we consider in Theorem 2 is the critical regime for the Steklov problem, where we observe a change of behaviour in the limiting problem
Summary
Let ⊂ Rd be a bounded and connected domain with Lipschitz boundary ∂. Is the Laplacian, and ∂ν is the outward pointing normal derivative. Both problems consist in finding the eigenvalues μ and σ such that there exist non-trivial smooth solutions to the boundary value problems (1) and (2). For both problems, the spectra form discrete unbounded sequences. 0 = σ0 < σ1 σ2 · · · ∞, where each eigenvalue is repeated according to multiplicity. The corresponding eigenfunctions { fk} and {uk} have natural normalisations as orthonormal bases of L2( ) and L2(∂ ), respectively
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