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Inverse stability for hyperbolic equations with different initial conditions

Shiqi Ma

TL;DR

This work addresses the stable recovery of a potential and initial data for a hyperbolic IBVP from a single boundary observation. It introduces two core tools—the weighted energy estimate and a novel pointwise Carleman estimate with explicit boundary terms—and then combines them to prove Lipschitz stability for recovering $(q,a,b)$, even when initial data differ between potentials under a pointwise positivity condition. It further studies the initial-potential problem, achieving Lipschitz stability for $q$ without the positivity constraint, highlighting a closer connection to fixed-angle inverse scattering. Overall, the authors develop a shorter, clearer Carleman-based framework that avoids time reflection and extends stability results to new coupling of initial data with the potential, with implications for inverse scattering problems.

Abstract

We establish Lipschitz stability for both the potential and the initial conditions from a single boundary measurement in the context of a hyperbolic boundary initial value problem. In our setting, the initial conditions are allowed to differ for different potentials. Compared to the traditional B-K method, our approach does not require the time reflection step. This advantage makes it possible to apply our method to the fixed angle inverse scattering problem, which remains unresolved for the single incident wave case. To achieve our result, we impose certain pointwise positivity assumption on the difference of initial conditions. The assumption generalizes previous stability results that usually assume the difference to be zero. We propose the initial-potential problem and prove a potential inverse stable recovery result of it. The initial-potential problem serves as an attempt to relate initial boundary value problem with the scattering problem, and to explore the possibility to relax the positivity requirement on the initial data. We also establish a new pointwise Carleman estimate, whose proof is significantly shorter and the reasoning is much clearer than traditional ones.

Inverse stability for hyperbolic equations with different initial conditions

TL;DR

This work addresses the stable recovery of a potential and initial data for a hyperbolic IBVP from a single boundary observation. It introduces two core tools—the weighted energy estimate and a novel pointwise Carleman estimate with explicit boundary terms—and then combines them to prove Lipschitz stability for recovering , even when initial data differ between potentials under a pointwise positivity condition. It further studies the initial-potential problem, achieving Lipschitz stability for without the positivity constraint, highlighting a closer connection to fixed-angle inverse scattering. Overall, the authors develop a shorter, clearer Carleman-based framework that avoids time reflection and extends stability results to new coupling of initial data with the potential, with implications for inverse scattering problems.

Abstract

We establish Lipschitz stability for both the potential and the initial conditions from a single boundary measurement in the context of a hyperbolic boundary initial value problem. In our setting, the initial conditions are allowed to differ for different potentials. Compared to the traditional B-K method, our approach does not require the time reflection step. This advantage makes it possible to apply our method to the fixed angle inverse scattering problem, which remains unresolved for the single incident wave case. To achieve our result, we impose certain pointwise positivity assumption on the difference of initial conditions. The assumption generalizes previous stability results that usually assume the difference to be zero. We propose the initial-potential problem and prove a potential inverse stable recovery result of it. The initial-potential problem serves as an attempt to relate initial boundary value problem with the scattering problem, and to explore the possibility to relax the positivity requirement on the initial data. We also establish a new pointwise Carleman estimate, whose proof is significantly shorter and the reasoning is much clearer than traditional ones.

Paper Structure

This paper contains 6 sections, 9 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Two estimates
  3. Weighted energy estimate
  4. Carleman estimate with boundary terms
  5. Combination of the two estimates
  6. proofs of the inverse stability

Key Result

Theorem 1.1

Let integer $s$ in eq:1con-PR25 satisfies $s \geq \lceil n/2 \rceil + 1$. Let $u_j$ satisfies eq:1-PR25-eq:1con-PR25 with $q_j$, $a_j$, $b_j$ and $g_j~(j=1,2)$, and the corresponding compatibility conditions are satisfied up to order $s$. Assume there are two positive constants $C_1$, $C_2$ such tha Under Assumption asm:assu-PR25, there exist two positive numbers $T$ and $C = (C_1, C_2, \widetilde

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.2
  • Proposition 2.1: Weighted energy estimate
  • proof
  • Lemma 2.2
  • proof
  • ...and 14 more