Unique continuation for the wave equation: the stability landscape
Abstract
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.
Keywords
Cite
@article{arxiv.2510.20359,
title = {Unique continuation for the wave equation: the stability landscape},
author = {Erik Burman and Lauri Oksanen and Janosch Preuss and Ziyao Zhao},
journal= {arXiv preprint arXiv:2510.20359},
year = {2025}
}
Comments
38 pages, 8 figures