From Steklov to Neumann via homogenisation
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From Steklov to Neumann via homogenisation Alexandre Girouard, Antoine Henrot & Jean Lagacé Communicated by G. Dal Maso
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 perimeternormalized 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.
1. Introduction Let ⊂ Rd be a bounded and connected domain with Lipschitz boundary ∂. Consider on the Neumann eigenvalue problem
− f = μf ∂ν f = 0
in , on ∂,
(1)
as well as the Steklov eigenvalue problem
u = 0 ∂ν u = σ u
in , on ∂.
(2)
A. Girouard et al.
Here 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 = μ0 < μ1 μ2 . . . ∞ and 0 = σ0 < σ1 σ2 · · · ∞, where each eigenvalue is repeated according to multiplicity. The corresponding eigenfunctions { f k } and {u k } have natural normalisations as orthonormal bases of L2 () and L2 (∂), respectively. 1.1. From Steklov to Neumann : heuristics 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. Indeed, in both cases the eigenvalues are those of a differential or pseudo-differential operator, namely the Laplacian and the Dirichletto-Neumann map, whose kernels consist of constant functions. Moreover, in both cases, the natural isoperimetric type problem consists in maximizing μk and σk (instead of minimizing it as is usual for the Dirichlet problem). The relation between the two boundary value problems is not solely heuristic and incidental. Indeed, it is known from the works of Arrieta–Jiménez-Casas–Rodriguez-Bernal [3] and Lamberti–Provenzano [30,31] that one can recover the Steklov problem as a limit of weighted Neumann problems − f = μρε f in , (3) on ∂, ∂ν f = 0 where ρε is a density function whose support converges to the boundary as ε → 0. If we are to interpret the Neumann problem as finding the frequencies and modes of vib
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