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Preprints, Working Papers, ... Year : 2023

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 Hörmander 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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Dates and versions

hal-04151529 , version 1 (05-07-2023)

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Camille Laurent, Matthieu Léautaud. Lectures on unique continuation for waves. 2023. ⟨hal-04151529⟩
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