A Steklov-spectral approach for solutions of Dirichlet and Robin boundary value problems

In this paper we revisit an approach pioneered by Auchmuty to approximate solutions of the Laplace- Robin boundary value problem. We demonstrate the efficacy of this approach on a large class of non-tensorial domains, in contrast with other spectral approaches for such problems. We establish a spectral approximation theorem showing an exponential fast numerical evaluation with regards to the number of Steklov eigenfunctions used, for smooth domains and smooth boundary data. A polynomial fast numerical evaluation is observed for either non-smooth domains or non-smooth boundary data. We additionally prove a new result on the regularity of the Steklov eigenfunctions, depending on the regularity of the domain boundary. We describe three numerical methods to compute Steklov eigenfunctions

Domains without dense Steklov nodal sets

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem −Δϕ_(σj) = 0, on Ω, ∂_νϕ_(σj) = σ_jϕ_(σj) on ∂Ω in two-dimensional domains Ω. In particular, this paper presents a dense family A of simply-connected two-dimensional domains with analytic boundaries such that, for each Ω∈A, the nodal set of the eigenfunction ϕ_(σj) “is not dense at scale σ_j⁻¹”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains Ω∈A, the nodal sets of the eigenfunctions ϕ_(σj) associated with the eigenvalue σ_j have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each Ω∈A there is a value r₁ > 0 such that for each j there is x_j ∈ Ω such that ϕ_(σj) does not vanish on the ball of radius r₁ around x_j

Funktionentheorie

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Variational aspects of Laplace eigenvalues on Riemannian surfaces

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of λk-extremal metrics and the existence of a partially regular λ₁-maximiser