Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Logunov, Alexander

doi:10.4007/annals.2018.187.1.4

Cited by 112 publications

(139 citation statements)

References 18 publications

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“…It was known to Agmon [2] and Almgen [4] that F h (x, r) is an increasing function of r and thus the function t → log H h (x, e t ) is convex; F h is called the frequency function of h. Garofalo and Lin [36] showed that a similar almost monotonicity inequality holds for solutions of second order elliptic PDEs in divergence form with Lipschitz coefficients, which has many applications to nodal sets on smooth manifolds. We omit the accurate definition of the frequency in that setting, but we would like to describe the relation of see [38], [42], [55].…”

Section: 2mentioning

confidence: 99%

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Logunov

Malinnikova

2018

Current Developments in Mathematics

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This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.

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Section: 2mentioning

confidence: 99%

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Logunov

Malinnikova

2018

Current Developments in Mathematics

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“…Classical results in this area may be found in [1,7,12,5,6,3]. See [13,14] and their references for more recent advances. In particular, for ε = 1, Theorem 1.1 was proved in [6] by Q. Han and F. Lin.…”

Section: Introductionmentioning

confidence: 98%

Nodal Sets and Doubling Conditions in Elliptic Homogenization

Lin

Shen

2019

Acta. Math. Sin.-English Ser.

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This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L ε } in divergence form with rapidly oscillating and periodic coefficients. We show that the (d − 1)-dimensional Hausdorff measures of the nodal sets of solutions to L ε (u ε ) = 0 in a ball in R d are bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u ε by solutions of the homogenized equation.

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“…for every j ≥ 1. Yau's conjecture was proved by Donnelly and Fefferman in [DF88] for real analytic manifolds, and the lower bound was established by Logunov and Malinnikova [Log16a,Log16b,LM16] for the general case.…”

Section: Introduction 1background and Motivationsmentioning

confidence: 95%

Random nodal lengths and Wiener chaos

Rossi¹

2019

Probabilistic Methods in Geometry, Topology and Spectral Theory

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In this survey we collect some of the recent results on the "nodal geometry" of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics). We give some insight on both Berry's cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic components. Finally we consider the general case of monochromatic random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian surface with frequencies from a short interval, whose scaling limit is Berry's Random Wave Model. For the latter we present some recent results on the asymptotic distribution of its nodal length in the high energy limit (equivalently, for growing domains).

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Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Abstract: Abstract. Let M be a compact C ∞ -smooth Riemannian manifold of dimension n, n ≥ 3, and let ϕ λ : ∆M ϕ λ + λϕ λ = 0 denote the Laplace eigenfunction on M corresponding to the eigenvalue λ. We show that

Cited by 112 publications

References 18 publications

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Nodal Sets and Doubling Conditions in Elliptic Homogenization

Random nodal lengths and Wiener chaos

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