On the Inner Radius of Nodal Domains
Abstract
Description
Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λ^k. 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.
Reference added, slight change in estimate in theorem 1, remark on the new proof in dim two added, to appear in Canad. Math. Bull
Reference added, slight change in estimate in theorem 1, remark on the new proof in dim two added, to appear in Canad. Math. Bull