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

2017/12/21 by Alexander Logunov · 9 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Nonlinear Partial Differential Equations

paper · doi:10.4007/annals.2018.187.1.4

Abstract

Let \mathbbM be a compact C^∞-smooth Riemannian manifold of dimension n, n≥ 3, and let φ_λ = Δ_Mφ_λ + λφ_λ = 0 denote the Laplace eigenfunction on \mathbbM corresponding to the eigenvalue λ. We show that H^n-1(\φ_λ = 0\) ≤ Cλ^α, where α > 1/2 is a constant, which depends on n only, and C>0 depends on \mathbbM. This result is a consequence of our study of zero sets of harmonic functions on C^∞-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.

Cited by

• Global controllability of the Cahn-Hilliard equation
• Lower Bound of Nodal Sets in Elliptic Homogenization and Functions with Strong Maximum Principle
• Quantitative unique continuation for Neumann problem in planar C^1,α domains
• Nodal Count for Orthogonally Invariant Ensembles
• Propagation of smallness near codimension two for gradients of harmonic functions
• Nadirashvili' Conjecture for Elliptic PDEs and its Applications
• Elliptic unique continuation below the Lipschitz threshold
• Nodal set for the Schrödinger equation under a local growth condition
• On the nodal domain count under metric perturbations, Courant sharpness and boundary intersections

Related

• The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions
• The size of the nodal sets for the eigenfunctions of the smooth laplacian
• Hausdorff measure bound for the nodal sets of Neumann Laplace eigenfunctions
• Critical set of eigenfunctions of the Laplacian
• The Sharp Measure Upper Bound of the Nodal Sets of Neumann Laplace Eigenfunctions on C1,1 Domains