On the blowup of quantitative unique continuation estimates for waves and applications to stability estimates

TL;DR

This paper investigates the blowup behavior of a geometric constant in quantitative unique continuation for wave equations, providing bounds and applications to stability estimates and inverse problems.

Contribution

It establishes an upper bound for the geometric constant as the domain shrinks, and applies this to derive stability estimates for unique continuation and hyperbolic inverse problems.

Findings

Bounded the blowup of the geometric constant as the domain shrinks to zero.

Derived stability estimates for unique continuation up to the maximal domain.

Extended results to hyperbolic inverse problems using Carleman estimates.

Abstract

In this paper we are interested in the blowup of a geometric constant $\mathfrak{C}(\delta)$ appearing in the optimal quantitative unique continuation property for wave operators. In a particular geometric context we prove an upper bound for $\mathfrak{C}(\delta)$ as $\delta$ goes to $0$. Here $\delta>0$ denotes the distance to the maximal unique continuation domain. As applications we obtain stability estimates for the unique continuation property up to the maximal domain. Using our abstract framework~\cite{FO25abstract} we also derive a stability estimate for a hyperbolic inverse problem. The proof is based on a global explicit Carleman estimate combined with the propagation techniques of Laurent-L\'eautaud.