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

Spyridon Filippas, Lauri Oksanen

TL;DR

The paper addresses quantitative unique continuation for the wave equation near the maximal domain of propagation by deriving an explicit global Carleman estimate with a Gaussian time weight and tracking the geometric dependence on the distance $\delta$ to the optimal cone. The authors establish a sharp exponential blow-up rate for the best possible observability constant $\mathfrak{C}(\delta)$ as $\delta\to 0$, and they prove stability estimates up to the maximal domain, with log-type bounds inside the diamond and log-log bounds up to the cone; these results are complemented by a local explicit quantitative estimate that is iterated along level sets to obtain global results. The approach blends a global Carleman framework, propagation techniques of Laurent–Léautaud, time-frequency localization, and complex-analytic tools to deliver explicit $\delta$-dependent constants. The findings have implications for inverse problems and controllability, providing precise cost-of-control and stability estimates that elucidate how ill-posedness deteriorates as one nears the threshold of unique continuation. The work also connects to the Boundary Control method, offering quantitative insight into stability in recovering coefficients or metrics from boundary data in geometric settings. All results are stated with explicit constants and are designed to be usable in applications requiring precise geometric dependence.

Abstract

In this paper we are interested in the blowup of a geometric constant $\mathfrak{C}(δ)$ appearing in the optimal quantitative unique continuation property for wave operators. In a particular geometric context we prove an upper bound for $\mathfrak{C}(δ)$ as $δ$ goes to $0$. Here $δ>0$ 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-Léautaud.

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

TL;DR

The paper addresses quantitative unique continuation for the wave equation near the maximal domain of propagation by deriving an explicit global Carleman estimate with a Gaussian time weight and tracking the geometric dependence on the distance to the optimal cone. The authors establish a sharp exponential blow-up rate for the best possible observability constant as , and they prove stability estimates up to the maximal domain, with log-type bounds inside the diamond and log-log bounds up to the cone; these results are complemented by a local explicit quantitative estimate that is iterated along level sets to obtain global results. The approach blends a global Carleman framework, propagation techniques of Laurent–Léautaud, time-frequency localization, and complex-analytic tools to deliver explicit -dependent constants. The findings have implications for inverse problems and controllability, providing precise cost-of-control and stability estimates that elucidate how ill-posedness deteriorates as one nears the threshold of unique continuation. The work also connects to the Boundary Control method, offering quantitative insight into stability in recovering coefficients or metrics from boundary data in geometric settings. All results are stated with explicit constants and are designed to be usable in applications requiring precise geometric dependence.

Abstract

In this paper we are interested in the blowup of a geometric constant appearing in the optimal quantitative unique continuation property for wave operators. In a particular geometric context we prove an upper bound for as goes to . Here 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-Léautaud.

Paper Structure

This paper contains 15 sections, 29 theorems, 224 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results and comments
  4. Idea of the proof and plan of the paper
  5. The global Carleman estimate
  6. A "Carleman equality"
  7. The subelliptic estimate
  8. The unperturbed operator
  9. Perturbation with the Gaussian weight
  10. Using the Gaussian weight
  11. Qualitative unique continuation as a consequence: an example
  12. The local explicit quantitative estimate
  13. Iteration and global estimates
  14. Applications
  15. Some lemmata

Key Result

Theorem 1.1

Consider $q \in L^\infty(\mathcal{D})$ with $\left\Vert q \right\Vert_{L^\infty(\mathcal{D})}^{} \leq M$. There exists $N >0$ depending on $r_0,n,M$ only, such that the following holds. Let $\delta>0$ small and define $\mathcal{D}_\delta= \mathcal{D} \cap \mathcal{M}_\delta$. Then for any $u \in H^1

Figures (2)

  • Figure 1: A solution of the wave equation that vanishes on $(-s,s)\times \omega$ has to vanish in the diamond $\mathcal{D}$. Theorem \ref{['thm_up_to_cone']} gives a stability estimate in the whole diamond in the case where $s=R/2$ and $\omega$ is a ball of radius $R$.
  • Figure 2: The level sets of the foliating function $\phi$, which approach the boundary of the cone $\{\phi>0\}$ as $\delta$ goes to $0$.

Theorems & Definitions (55)

  • Theorem 1.1: Optimal stability estimate arbitrarily close to the maximal domain
  • Theorem 1.2: Stability estimate up to the optimal domain
  • Remark 1.3
  • Theorem 1.4: Theorem 1.2 in FO25abstract
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • ...and 45 more