L2 Bounds for Normal Derivatives of Dirichlet Eigenfunctions
Andrew Hassell, Terence C. Tao · 2003
Abstract. Suppose that M is a compact Riemannian manifold with boundary and u is an L 2-normalized Dirichlet eigenfunction with eigenvalue λ. Let ψ be its normal derivative at the boundary. Scaling considerations lead one to expect that the L 2 norm of ψ will grow as λ 1/2 as λ → ∞. We sketch proofs of an upper bound of the form �ψ � 2 2 ≤ Cλ for any Riemannian manifold, and a lower bound cλ ≤ �ψ � 2 2 provided that M has no trapped geodesics (see the main Theorem for a precise statement). Here c and C are positive constants that depend on M, but not on λ. Full details will appear in [3]. 1.