Small-scale mass estimates for Neumann eigenfunctions: piecewise smooth planar domains

2023/09/19 by Hans Christianson, John A. Toth, Christianson, Hans +1
Computer Science · Mathematics · #35B30 #35B40 #35G15 #35P99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2309.10875

openalex publication_date 2023/09/19 · openalex created_date 2023/09/22 · openalex updated_date 2026/07/28

Abstract

Let Ω be a piecewise-smooth, bounded convex domain in \R² and consider L²-normalized Neumann eigenfunctions ϕ_λ with eigenvalue λ². Our main result is a small-scale \em non-concentration estimate: We prove that for \em any x₀ ∈ Ω, (including boundary and corner points) and any δ∈ [0,1), ‖ ϕ_λ‖_{B(x0,λ^-δ)∩ Ω} = O(λ^-δ/2). The proof is a stationary vector field argument combined with a small scale induction argument.

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