Unique Continuation from Infinity in Asymptotically Anti-de Sitter Spacetimes
Unique Continuation from Infinity in Asymptotically Anti-de Sitter Spacetimes
复制标题
渐进反德西特时空中无穷大的唯一延拓
DOI：
10.1080/03605302.2017.1390677
复制
发表时间：
2015
影响因子：
2.4
通讯作者：
Arick Shao
中科院分区：
文献类型：
--
作者：
G. Holzegel;Arick Shao
通过研究Klein-Gordon型方程$${Box_g\Phi+\sigma\Phi={数学{G}}(\Phi，\Partial\Phi)}$$-gϕ+σϕ=G(ϕ，∂ϕ)，$${σ∈\in{\mathbb{R}}$$σ∈R，研究了渐近反de Sitter时空的唯一延拓性质.我们的主要结果证明，如果$$\ϕ在保形边界$${\mathcal{i}}$$i上在足够长的时间间隔内消失到足够高的阶(取决于$$\sigma}$$σ)，则解必然在$${\mathcal{i}}$$i的邻域内消失。特别地，在$${\sigma}$σ-范围内，对于正问题，当Dirichlet和Neumann条件都可能时，证明了这两个条件都是唯一的。时间间隔的长度可能与零测地线在这些背景上的重新聚焦时间有关，预计将是尖锐的。文中还讨论了引力摄动的一些全局应用和唯一性结果。这一证明是基于在这种情况下建立的新的Carleman估计。
We consider the unique continuation properties of asymptotically anti-de Sitter spacetimes by studying Klein–Gordon-type equations $${\Box_g \phi + \sigma \phi = {\mathcal{G}} ( \phi, \partial \phi )}$$□gϕ+σϕ=G(ϕ,∂ϕ), $${\sigma \in {\mathbb{R}}}$$σ∈R, on a large class of such spacetimes. Our main result establishes that if $${\phi}$$ϕ vanishes to sufficiently high order (depending on $${\sigma}$$σ) on a sufficiently long time interval along the conformal boundary $${{\mathcal{I}}}$$I, then the solution necessarily vanishes in a neighborhood of $${{\mathcal{I}}}$$I. In particular, in the $${\sigma}$$σ-range where Dirichlet and Neumann conditions are possible on $${{\mathcal{I}}}$$I for the forward problem, we prove uniqueness if both these conditions are imposed. The length of the time interval can be related to the refocusing time of null geodesics on these backgrounds and is expected to be sharp. Some global applications as well as a uniqueness result for gravitational perturbations are also discussed. The proof is based on novel Carleman estimates established in this setting.