Year: 2023
Type: preprint
Abstract: Let $Ω$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$. Our main result is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary and corner points) and any $δ\in [0,1),$ $$ \| ϕ_λ\|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ The proof is a stationary vector field argument combined with a small scale induction argument.
Source: arXiv (Cornell University)
Authors: Christianson, Hans, Toth, John A.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4386942658