The wave equation on asymptotically Anti-de Sitter spaces
arXiv:0911.5440
Abstract
In this paper we describe the behavior of solutions of the Klein-Gordon equation, (Box_g+lambda)u=f, on Lorentzian manifolds (X^o,g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the sense that the metric is conformal to a smooth Lorentzian metric g^ on X, where C has a non-trivial boundary, in the sense that g=x^{-2}g^, with x a boundary defining function. The boundary is conformally time-like for these spaces, unlike asymptotically de Sitter spaces, which are similar but with the boundary being conformally space-like. Here we show local well-posedness for the Klein-Gordon equation, and also global well-posedness under global assumptions on the (null)bicharacteristic flow, for lambda below the Breitenlohner-Freedman bound, (n-1)^2/4. These have been known under additional assumptions, see the work of Breitenlohner-Freedman and Holzegel. Further, we describe the propagation of singularities of solutions and obtain the asymptotic behavior (at the boundary of X) of regular solutions. We also define the scattering operator, which in this case is an analogue of the hyperbolic Dirichlet-to-Neumann map. Thus, it is shown that below the Breitenlohner-Freedman bound, the Klein-Gordon equation behaves much like it would for the conformally related metric, g^, with Dirichlet boundary conditions, for which propagation of singularities was shown by Melrose, Sjöstrand and Taylor, though the precise form of the asymptotics is different.
63 pages, 1 figure Updated version to appear in Analysis & PDE. In addition to various minor changes, Section 5 now has a brief discussion of b-pseudodifferential operators, the discussion of GBB in Section 6 has been extended (with some typos fixed), a sketch of the Melrose-Sjöstrand-Lebeau argument in Theorem 8.8 has been added, and a sign error in the statement of Proposition 8.6 is fixed