FRG Collaborative Proposal: Eigenfunctions of the Laplacian

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Collaborative proposal DMS 0354386/0354539/0354668 PIs: C.Sogge, S.Zelditch (Johns Hopkins)/D.Tataru, M.Zworski (U California, Berkeley)/H.Smith (U Washington) Title: Eigenfunctions of the Laplacian ABSTRACT This proposal is concerned with estimates of solutions of wave equations on (both compact and non-compact) Riemannian manifolds, possibly with boundary. We are interested in how the geometry, the boundary and the regularity of the metric influence certain basic estimates. Problems of this kind arise both in the study of local and global existence of solutions of nonlinear wave equations and in the study of eigenfunctions of Laplacians in quantum chaos. Although these topics are widely separated in their physical origins, the relevant mathematics is closely related and the techniques and insights in the two areas cross fertilize in a fruitful way. Each of the PIs is an expert one of these areas. By pooling our knowledge through a FRG grant we shall be able to make significant progress in these fields as well as possibly in related ones such as general relativity theory. Eigenfunctions are the building blocks of functions. Understanding their basic size and concentration properties helps us solve new differential equations. In the other direction, techniques from the study of nonlinear wave equations have recently been used to prove new results about eigenfunctions. This proposal is about continuing this cross-fertilization of related fields.