A natural lower bound for the size of nodal sets

Hezari, Hamid; Sogge, Christopher

doi:10.2140/apde.2012.5.1133

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A natural lower bound for the size of nodal sets

Hamid Hezari and Christopher D. Sogge

Vol. 5 (2012), No. 5, 1133–1137

DOI: 10.2140/apde.2012.5.1133

Abstract

We prove that, for an $n$ -dimensional compact Riemannian manifold $(M, g)$ , the $(n - 1)$ -dimensional Hausdorff measure $| Z_{λ} |$ of the zero-set $Z_{λ}$ of an eigenfunction $e_{λ}$ of the Laplacian having eigenvalue $- λ$ , where $λ \geq 1$ , and normalized by $\int_{M} | e_{λ} |^{2} d V_{g} = 1$ satisfies

C | Z_{λ} | \geq λ^{\frac{1}{2}} {(\int_{M} | e_{λ} | d V_{g})}^{2}

for some uniform constant $C$ . As a consequence, we recover the lower bound $| Z_{λ} | ≳ λ^{(3 - n) ∕ 4}$ .

Keywords

eigenfunctions, nodal lines

Mathematical Subject Classification 2010

Primary: 35P15

Milestones

Received: 12 August 2011

Accepted: 24 October 2011

Published: 29 December 2012

Authors

Hamid Hezari
Department of Mathematics University of California Irvine, CA 92697 United States

Christopher D. Sogge
Department of Mathematics Johns Hopkins University Baltimore, MD 21093 United States

Vol. 5, No. 5, 2012