Mathematical Research Letters

Volume 18 (2011)

Number 2

A geometric covering lemma and nodal sets of eigenfunctions

Pages: 337 – 352



Xiaolong Han (Wayne State University, Detroit, MI 48202, USA)

Guozhen Lu (Beijing Normal University, Beijing, China 100875 and Wayne State University, Detroit, MI 48202, USA)


The main purpose of this paper is two-fold. On one hand, we prove a sharper covering lemma in Euclidean space $\mathbb R^n$ for all $n\ge2$ (see Theorem \ref{newcovering}). On the other hand, we apply this covering lemma to improve existing results for BMO and volume estimates of nodal sets for eigenfunctions $u$ satisfying $\bigtriangleup u+\lambda u=0$ on $n$-dimensional Riemannian manifolds when $\lambda$ is large (see Theorems \ref{BMO1}, \ref{volume}). We also improve the BMO estimates for the function $q=|\nabla u|^2+\frac{\lambda}{n}u^2$ (see Theorem \ref{BMO2}). Our covering lemma sharpens substantially earlier results and is fairly close to the optimal one we can expect (Conjecture \ref{conjecture}).

Published 20 May 2011