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# On the Inner Radius of a Nodal Domain

Published:2008-06-01
Printed: Jun 2008
• Dan Mangoubi
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## Abstract

Let $M$ be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$. We give upper and lower bounds on the inner radius of the type $C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincar\'{e} type inequality proved by Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.
 MSC Classifications: 58J50 - Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 35P15 - Estimation of eigenvalues, upper and lower bounds 35P20 - Asymptotic distribution of eigenvalues and eigenfunctions