Lectures on unique continuation for waves

TL;DR

This paper introduces the concept of unique continuation for wave equations, proving key theorems under specific conditions, and discusses applications like controllability and related estimates.

Contribution

It presents new proofs of classical and modern unique continuation theorems for wave operators, including the Tataru theorem, under various conditions.

Findings

Proof of Hörmander's local unique continuation theorem under pseudoconvexity.

Proof of Tataru's theorem for wave operators with time-independent coefficients.

Application to approximate controllability and related quantitative estimates.

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.