Weyl's law for the Steklov problem on surfaces with rough boundary

Abstract

The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundaries is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl's law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as "slow" exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings.

Flexibility of Steklov eigenvalues via boundary homogenisation

“…We study the weighted Steklov problem 1.1 through the formalism developed in [10], see also [13,16]. For any domain with Lipschitz boundary ⊂ M and any Radon measure μ supported on , we define the Sobolev spaces W 1, p ( , μ) as the closure of C ∞ ( ) under the norm…”

“…By Koebe's uniformisation theorem [15], there exists a circle domain ⊂ R 2 (i.e. a domain whose boundary is disjoint union of circles) and a conformal diffeomorphism ϕ : → 0,b such that g 0 = ϕ * g. It follows from [13,Theorem 1.6] that σ k ( 0,b , g) = σ k ( , g 0 , |dϕ|) for every k ∈ N. Furthermore, it follows from the proof of [2, Lemma 5.1] that there is p > 1 so that |dϕ| ∈ L p (∂ ) . Therefore, by Theorem 1.5 there exists a sequence of domains ε ⊂ with the same topological type so that σ k ( ε , g 0 ) → σ k ( , g 0 , |dϕ|) as ε → 0, concluding the proof.…”