On the Inner Radius of Nodal Domains

Details On the Inner Radius of Nodal Domains On the Inner Radius of Nodal Domains

2005-11-14

arxiv.org/abs/math/0511329v3

Canad. Math. Bull. 51 (2008), no. 2, 249--260

Mathematics

Spectral Theory

Reference added, slight change in estimate in theorem 1, remark on the new proof in dim two added, to appear in Canad. Math. B

Scientific paper

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^k. 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.

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