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arXiv · 2307.02155

Lectures on unique continuation for waves

Camille Laurent·Matthieu Léautaud
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Abstract

These notes are intended as an introduction to the question of unique continuation for the wave operator, and some of its applications. The general question is whether a solution to a wave equation in a domain, vanishing on a subdomain has to vanish everywhere. We state and prove two of the main results in the field. We first give a proof of the classical local H{\"o}rmander theorem in this context which holds under a pseudoconvexity condition. We then specialize to the case of wave operators with time-independent coefficients and prove the Tataru theorem: local unique continuation holds across any non-characteristic hypersurface. This local result implies a global unique continuation statement which can be interpreted as a converse to finite propagation speed. We finally give an application to approximate controllability, and present without proofs the associated quantitative estimates.

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Camille Laurent, Matthieu Léautaud. 2023-07-05. Lectures on unique continuation for waves. https://arxiv.org/abs/2307.02155

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