论文标题
无效的大地测量并改善了在渐近抗DE的空间中的波浪的独特延续
Null Geodesics and Improved Unique Continuation for Waves in Asymptotically Anti-de Sitter Spacetimes
论文作者
论文摘要
我们考虑了克莱因（Klein） - 戈登方程的解决方案是否可以从共同边界中继续进行。正面答案首先由G. holzegel的第二作者在边界几何形状的适当假设下给出，并在足够长的时间内施加的边界数据给出了正面答案。关键步骤是建立Carleman估计的Klein-Gordon操作员，附近的连接边界附近。 在本文中，我们进一步改善了上述结果。首先，我们建立了新的Carleman估计值---因此，对于Klein-Gordon方程，新的独特的延续结果，尤其是更一般的边界几何形状。其次，我们通过将它们连接到保形边界附近的空测量学的轨迹来提出我们假设的最佳性。这些大地测量学在对独特延续的反例构建中起着至关重要的作用。最后，我们开发了一种新的协变形式主义，该形式主义将在本文中有用，无论是目前，更普遍的范围 - 用于治疗在保形边界处具有渐近限制的紧张对象。
We consider the question of whether solutions of Klein--Gordon equations on asymptotically Anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive answers were first given by the second author with G. Holzegel, under suitable assumptions on the boundary geometry and with boundary data imposed over a sufficiently long timespan. The key step was to establish Carleman estimates for Klein--Gordon operators near the conformal boundary. In this article, we further improve upon the above-mentioned results. First, we establish new Carleman estimates---and hence new unique continuation results---for Klein--Gordon equations on a larger class of spacetimes, in particular with more general boundary geometries. Second, we argue for the optimality, in many respects, of our assumptions by connecting them to trajectories of null geodesics near the conformal boundary; these geodesics play a crucial role in the construction of counterexamples to unique continuation. Finally, we develop a new covariant formalism that will be useful---both presently and more generally beyond this article---for treating tensorial objects with asymptotic limits at the conformal boundary.
