On the Inner Radius of a Nodal Domain

Mangoubi, Dan

Canadian mathematical bulletin, Tome 51 (2008) no. 2, pp. 249-260

Résumé

Let $M$ be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue $\text{ }\!\!\lambda\!\!\text{ }.$ 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 Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

Détail

DOI : 10.4153/CMB-2008-026-2

Mots-clés : 58J50, 35P15, 35P20

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