We consider a unique continuation problem for the wave equation given data in a volumetric subset of the space time domain. In the absence of data on the lateral boundary of the space-time cylinder we prove that the solution can be continued with H\"older stability into a certain proper subset of the space-time domain. Additionally, we show that unique continuation of the solution to the entire space-time cylinder with Lipschitz stability is possible given the knowledge of a suitable finite dimensional space in which the trace of the solution on the lateral boundary is contained. These results allow us to design a finite element method that provably converges to the exact solution at a rate that mirrors the stability properties of the continuous problem.

翻译：我们考虑波动方程在时空域的一个体积子集上给定数据的唯一延拓问题。在缺乏时空柱体侧边界数据的情况下，我们证明解能以Hölder稳定性延拓到时空中某个特定真子集内。此外，我们证明：若已知侧边界上解的迹包含于某个合适的有限维空间，则解能以Lipschitz稳定性唯一延拓至整个时空柱体。这些结果使我们能够设计一种有限元方法，其收敛到精确解的速率严格对应连续问题稳定性特征。