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

Let $\Omega \subset \mathbb{R}^3$ be a bounded, convex domain with piecewise-smooth boundary, and consider $L^2$-normalized Neumann eigenfunctions $u$ satisfying$-h^2\Delta u = u$.Our main result is a small-scale non-concentration estimate: For any $p \in \bar{\Omega}$ (including boundary and edge points) and any $0 \leq \theta < 1$,\[\|u\|^2_{L^2(\Omega \cap B(p, h^{\theta}))} = O(h^{\theta}).\]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 \cite{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.