Small-Scale Mass Estimates for Neumann Eigenfunctions: 3D Convex Domains with Planar Edges

Let Ω ⊂ R³ be a bounded, convex domain with piecewise-smooth boundary, and consider L²-normalized Neumann eigenfunctions u satisfying-h²Δ u = u.Our main result is a small-scale non-concentration estimate: For any p ∈ Ω̄ (including boundary and edge points) and any 0 ≤ θ < 1,\[\|u\|^2_{L^2(Ω ∩ B(p, hθ))} = O(hθ).\]We prove this for interior points, for boundary points on the convex domain, and for edge points where a smooth boundary surface meets a planar face. This provides the first three-dimensional extension of the two-dimensional result by Christianson and Toth {ChristiansonToth2020}, using many of the same techniques, including a stationary vector field commutator argument combined with a small-scale induction on h. For completeness, we also include the analogous statement and proof for Dirichlet boundary conditions.