, Volume 321, Issue 1, pp 85-111
Date: 03 May 2013

The Massive Wave Equation in Asymptotically AdS Spacetimes

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Abstract

We consider the massive wave equation on asymptotically AdS spaces. We show that the timelike \({\fancyscript{F}}\) behaves like a finite timelike boundary, on which one may impose the equivalent of Dirichlet, Neumann or Robin conditions for a range of (negative) mass parameter which includes the conformally coupled case. We demonstrate well posedness for the associated initial-boundary value problems at the H 1 level of regularity. We also prove that higher regularity may be obtained, together with an asymptotic expansion for the field near \({\fancyscript{F}}\) . The proofs rely on energy methods, tailored to the modified energy introduced by Breitenlohner and Freedman. We do not assume the spacetime is stationary, nor that the wave equation separates.

Communicated by P.T. Chruściel