Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains

Lagacé, Jean

doi:10.4153/s0008414x19000130

Cited by 6 publications

(3 citation statements)

References 25 publications

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“…The interested reader is instead referred to [47] and references therein. Problems concerning the asymptotic behaviour of solutions of spectral shape optimization problems saw a rise in interest in recent years, largely motivated by a connection to the famous Pólya conjecture highlighted in [20] (see also [36]); see, for instance, [3,16,9,10,11,41,35,65,60,17] where a variety of problems of this type are studied. Some elements of our proofs.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

Frank

Larson

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

Frank

Larson

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…There are also situations where lower bounds for k can be transferred to k . For instance, restricting to flat metrics on T 2 , it follows from Kao-Lai-Osting [43] and Lagacé [58] that…”

Section: Lower Bounds and Exact Values For Kmentioning

confidence: 99%

Large Steklov eigenvalues via homogenisation on manifolds

Girouard

Lagacé

2021

Invent. math.

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Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of $$8\pi $$ 8 π for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than $$2\pi $$ 2 π . This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

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“…The analysis of high energy eigenfunctions on compact Riemannian manifolds in general, and on flat tori in particular [12,1,13,17] is a central topic in geometric analysis where techniques from a number of areas of mathematics come into play.…”

Section: Introductionmentioning

confidence: 99%

Localization properties of high energy eigenfunctions on flat tori

Enciso¹,

García-Ruiz²,

Peralta‐Salas³

2021

Preprint

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We consider the question of when the Laplace eigenfunctions on an arbitrary flat torus T Γ := R d /Γ are flexible enough to approximate, over the natural length scale of order 1/ √ λ where λ ≫ 1 is the eigenvalue, an arbitary solution of the Helmholtz equation ∆h + h = 0 on R d . This problem is motivated by the fact that, by the asymptotics for the local Weyl law, "approximate Laplace eigenfunctions" do have this approximation property on any compact Riemannian manifold. What we find is that the answer depends solely on the arithmetic properties of the spectrum. Specifically, recall that the eigenvalues of T Γ are of the form λ k = Q Γ (k), where Q Γ is a quadratic form and k ∈ Z d . Our main result is that the eigenfunctions of T Γ have the desired approximation property if and only Q Γ is a multiple of a quadratic form with integer coefficients. In particular, the set of lattices Γ for which this approximation property holds has measure zero but includes all rational lattices. A consequence of this fact is that when Q Γ is a multiple of a quadratic form with integer coefficients, Laplace eigenfunctions exhibit an extremely flexible behavior over scales of order 1/ √ λ. In particular, there are eigenfunctions of arbitrarily high energy that exhibit nodal components diffeomorphic to any compact hypersurface of diameter O(1/ √ λ).

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Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains

Cited by 6 publications

References 25 publications

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

Large Steklov eigenvalues via homogenisation on manifolds

Localization properties of high energy eigenfunctions on flat tori

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