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Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space

Published version
Peer-reviewed

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Authors

Holzegel, Gustav 
Luk, Jonathan 
Smulevici, Jacques 

Abstract

Abstract: We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

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Article

Journal Title

Communications in Mathematical Physics

Conference Name

Journal ISSN

0010-3616
1432-0916

Volume Title

374

Publisher

Springer Berlin Heidelberg
Sponsorship
European Research Council (337488)
National Science Foundation (DMS-1204493)