Sharp Strichartz estimates on nontrapping asymptotically conic manifolds

Abstract

We obtain the Strichartz inequalities ║u║ L q t L r x ([0,1]× M ) ≥ C║ u (0) L 2 ║( M ) for any smooth n -dimensional Riemannian manifold M which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and nontrapping, where u is a solution to the Schrödinger equation iu t + 1/2 Δ M u = 0, and 2 < q, r ≥ ∞ are admissible Strichartz exponents (2/ q + n/r = n /2). This corresponds with the estimates available for Euclidean space (except for the endpoint ( q, r ) = (2, 2 n/n -2) when n > 2). These estimates imply existence theorems for semi-linear Schrödinger equations on M , by adapting arguments from Cazenave and Weissler and Kato. This result improves on our previous result, which was an L 4 t,x Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically flat manifolds.

Locations

• American Journal of Mathematics - View
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View - PDF
• American Journal of Mathematics - View
• arXiv (Cornell University) - PDF
• arXiv (Cornell University) - View - PDF