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arxiv: 2601.00648 · v3 · pith:QIQVK5QMnew · submitted 2026-01-02 · 🧮 math.AP

Lipschitz Stability for an Inverse Problem of Biharmonic Wave Equations with Damping

Minghui Bi , Yixian Gao This is my paper

Pith reviewed 2026-05-21 16:07 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse problembiharmonic wave equationLipschitz stabilitydampingobservability inequalitymultiplier methoddensity coefficientinitial displacement
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The pith

Lipschitz stability holds for simultaneous recovery of density coefficient and initial displacement in a damped biharmonic wave equation from boundary Laplacian data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that a variable density coefficient and the initial displacement can be recovered with Lipschitz stability from boundary measurements of the Laplacian of the solution and its normal derivative in a damped biharmonic wave equation. The authors first prove that the system operator generates a contraction semigroup to secure well-posedness of the forward problem. They then apply multiplier techniques to obtain a key observability inequality, from which explicit stability estimates follow. These estimates show that the biharmonic structure improves stability, with constants depending explicitly on the damping coefficient through the factor square root of one plus gamma. A reader would care because the result supplies a rigorous basis for parameter identification in applications such as non-destructive testing.

Core claim

The paper claims that the inverse problem for the damped biharmonic wave equation admits Lipschitz stability estimates for the simultaneous recovery of the variable density and initial displacement from the boundary Cauchy data consisting of Delta u and partial_n(Delta u) on the boundary. This follows from first showing that the system operator generates a contraction semigroup and then deriving an observability inequality via multiplier techniques, yielding stability constants with explicit dependence on the damping coefficient gamma via the factor (1 + gamma)^{1/2}.

What carries the argument

The observability inequality obtained via multiplier techniques for the damped biharmonic system, which directly produces the Lipschitz stability estimates.

If this is right

  • The biharmonic structure inherently enhances the stability of parameter identification relative to lower-order equations.
  • The stability constants grow explicitly with the damping coefficient according to the factor (1 + gamma)^{1/2}.
  • Well-posedness of the forward problem holds because the system operator generates a contraction semigroup.
  • The estimates provide a theoretical foundation for applications in non-destructive testing and dynamic inversion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The multiplier-based observability approach may carry over to inverse problems for other higher-order plate or beam equations.
  • Numerical schemes for the inverse problem could be designed to achieve the predicted Lipschitz rates in concrete geometries.
  • Similar explicit dependence on damping might appear when recovering coefficients in related damped hyperbolic systems.

Load-bearing premise

Multiplier techniques succeed in producing a sufficiently strong observability inequality from the given boundary Cauchy data under the assumed regularity and compatibility conditions on the domain and coefficients.

What would settle it

For a fixed domain, damping value, and two different densities or initial displacements, measure the difference in the boundary data and check whether the difference in the recovered quantities exceeds the claimed Lipschitz constant times that data difference.

read the original abstract

This paper establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient and the initial displacement in a damped biharmonic wave equation. The data consist of the boundary Cauchy data for the Laplacian of the solution, \(\Delta u |_{\partial \Omega}\) and \( \partial_{n}(\Delta u)|_{\partial \Omega}.\) We first prove that the associated system operator generates a contraction semigroup, which ensures the well-posedness of the forward problem. A key observability inequality is then derived via multiplier techniques. Building on this foundation, explicit stability estimates for the inverse problem are obtained. These estimates demonstrate that the biharmonic structure inherently enhances the stability of parameter identification, with the stability constants exhibiting an explicit dependence on the damping coefficient via the factor \( (1 + \gamma)^{1/2} \). This work provides a rigorous theoretical basis for applications in non-destructive testing and dynamic inversion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient ρ(x) and the initial displacement from boundary Cauchy data Δu|∂Ω and ∂n(Δu)|∂Ω in a damped biharmonic wave equation. The proof first shows that the system operator generates a contraction semigroup to establish well-posedness of the forward problem, then derives a key observability inequality via multiplier techniques, and finally obtains explicit stability estimates whose constants depend on the damping coefficient through the factor (1 + γ)^{1/2}.

Significance. If the observability inequality is valid under the paper's assumptions, the result supplies explicit Lipschitz constants for an inverse problem involving a fourth-order hyperbolic operator, which is relevant to plate models in elasticity and non-destructive testing. The explicit dependence on γ and the emphasis on the biharmonic structure providing enhanced stability constitute clear strengths of the analysis.

major comments (1)
  1. [§4] §4 (Observability inequality derivation): the multiplier method applied to the variable-coefficient biharmonic operator produces commutator terms containing ∇ρ and second derivatives of u; these must be absorbed by the damping term γ∂tu and the given boundary traces. The manuscript does not explicitly impose a geometric control condition on the bicharacteristics or assume that Ω is star-shaped, so it is unclear whether the remainder terms remain controlled uniformly for arbitrary smooth ρ, which directly affects whether the Lipschitz constant stays finite.
minor comments (2)
  1. [Abstract] The abstract states that the biharmonic structure 'inherently enhances the stability' but does not indicate which specific feature of the fourth-order operator (e.g., the extra boundary trace) is responsible for the improvement over the second-order case.
  2. [Introduction] Notation for the boundary operators Δu|∂Ω and ∂n(Δu)|∂Ω should be introduced once in the introduction and used consistently thereafter to avoid minor ambiguity in the data description.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the detailed comments, which help improve the clarity of the manuscript. We address the major comment point by point below.

read point-by-point responses

  1. Referee: §4 (Observability inequality derivation): the multiplier method applied to the variable-coefficient biharmonic operator produces commutator terms containing ∇ρ and second derivatives of u; these must be absorbed by the damping term γ∂tu and the given boundary traces. The manuscript does not explicitly impose a geometric control condition on the bicharacteristics or assume that Ω is star-shaped, so it is unclear whether the remainder terms remain controlled uniformly for arbitrary smooth ρ, which directly affects whether the Lipschitz constant stays finite.

    Authors: We thank the referee for highlighting this important aspect of the proof. In Section 4, the observability inequality is derived by applying multipliers to the damped biharmonic equation. The commutator terms involving ∇ρ and higher derivatives of u are handled by integrating by parts and using the positivity of the damping coefficient γ. Specifically, the term γ ∂_t u provides a dissipative effect that absorbs the lower-order commutators, while the boundary measurements of Δu and ∂_n(Δu) control the boundary contributions. Although no explicit geometric control condition is stated, the analysis relies on the smoothness of the coefficients and the boundedness of Ω to ensure uniform bounds. The resulting Lipschitz constant's dependence on (1 + γ)^{1/2} arises precisely from this absorption process. To address the concern, we will revise the manuscript to include a more detailed estimate of the commutator terms and clarify the assumptions on Ω and ρ in the revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent semigroup and multiplier arguments

full rationale

The paper first establishes well-posedness by showing the system operator generates a contraction semigroup, then derives an observability inequality via multiplier techniques applied to the damped biharmonic equation, and finally constructs explicit Lipschitz stability estimates from that inequality. These steps form a standard forward-to-inverse chain in which the observability inequality is obtained from the PDE and boundary data rather than being presupposed or fitted to the target recovery result. The factor (1 + γ)^{1/2} enters as an explicit model parameter in the estimates, not as a fitted quantity renamed as a prediction. No self-citations, ansatzes smuggled via prior work, or self-definitional reductions are present in the described derivation. The chain remains self-contained against external mathematical benchmarks such as semigroup theory and multiplier identities.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the paper relies on standard PDE well-posedness results and multiplier methods, with damping as a model parameter and the biharmonic operator as the core structure.

free parameters (1)
  • damping coefficient γ
    Appears explicitly in the stability constant (1 + γ)^{1/2}; treated as a given positive parameter in the model.
axioms (2)
  • domain assumption The associated system operator generates a contraction semigroup
    Invoked to ensure well-posedness of the forward problem as the first step in the abstract.
  • domain assumption Multiplier techniques produce a key observability inequality
    Used as the foundation for deriving the explicit stability estimates from boundary data.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Acosta and B

    S. Acosta and B. Palacios. Simultaneous determination of wave speed, diffusivity and nonlinearity in the westervelt equation using complex time-periodic solutions. arXiv:2509.10718, 2025

    work page arXiv 2025

  2. [2]

    I. S. Ar ˇzanyh. Quasianalytic solutions of the biharmonic equation. InDirect and inverse problems for partial differential equations and their applications (Russian), pages 55–61, 185. Izdat. “Fan” Uzbek. SSR, Tashkent, 1978

    work page 1978

  3. [3]

    M. A. Atahod ˇzaev. The exterior inverse biharmonic potential problem for a nearly spherical body. InBoundary value problems for differential equations, 4 (Russian), pages 77–93, 191. Izdat. “Fan” Uzbek. SSR, Tashkent, 1974

    work page 1974

  4. [4]

    Benrabah and N

    A. Benrabah and N. Boussetila. Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation.Inverse Probl. Sci. Eng., 27(3):340–368, 2019

    work page 2019

  5. [5]

    Bhattacharyya, K

    S. Bhattacharyya, K. Krupchyk, S. K. Sahoo, and G. Uhlmann. Inverse problems for third-order nonlinear pertur- bations of biharmonic operators.Comm. Partial Differential Equations, 50(3):407–440, 2025

    work page 2025

  6. [6]

    D. G. Bourgin. The Dirichlet problem for the damped wave equation.Duke Math. J., 7:97–120, 1940

    work page 1940

  7. [7]

    Bouslah, A

    Z. Bouslah, A. Hadj, and H. Saker. Shape reconstruction for an inverse biharmonic problem from partial Cauchy data.Math. Methods Appl. Sci., 47(3):1613–1627, 2024

    work page 2024

  8. [8]

    Cazenave and A

    T. Cazenave and A. Haraux.An introduction to semilinear evolution equations, volume 13 ofOxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998

    work page 1998

  9. [9]

    E. E. Chitorkin and N. P. Bondarenko. Inverse Sturm-Liouville problem with singular potential and spectral pa- rameter in the boundary conditions.J. Differential Equations, 421:495–523, 2025

    work page 2025

  10. [10]

    H. Q. N. Danh, D. O’Regan, V . A. V o, B. T. Tran, and C. H. Nguyen. Regularization of an initial inverse problem for a biharmonic equation.Adv. Difference Equ., pages Paper No. 255, 20, 2019

    work page 2019

  11. [11]

    Feizmohammadi

    A. Feizmohammadi. Reconstruction of 1D evolution equations and their initial data from one passive measure- ment.SIAM J. Math. Anal., 57(5):5089–5106, 2025

    work page 2025

  12. [12]

    F. G. Friedlander. Simple progressive solutions of the wave equation.Proc. Cambridge Philos. Soc., 43:360–373, 1947

    work page 1947

  13. [13]

    Y . Gao, H. Liu, and Y . Liu. On an inverse problem for the plate equation with passive measurement.SIAM J. Appl. Math., 83(3):1196–1214, 2023

    work page 2023

  14. [14]

    Hartman and A

    P. Hartman and A. Wintner. A criterion for the non-degeneracy of the wave equation.Amer. J. Math., 71:206–213, 1949

    work page 1949

  15. [15]

    Uniqueness and stability in determining the wave equation from a single passive boundary measurement, 2025

    Y . Kian and H. Liu. Uniqueness and stability in determining the wave equation from a single passive boundary measurement. arXiv:2507.10012, 2025

    work page arXiv 2025

  16. [16]

    Kian and F

    Y . Kian and F. Triki. Recovery of an inclusion in photoacoustic imaging.Inverse Probl. Imaging, 19(4):693–714, 2025

    work page 2025

  17. [17]

    Li and X

    P. Li and X. Wang. An inverse random source problem for the biharmonic wave equation.SIAM/ASA J. Uncertain. Quantif., 10(3):949–974, 2022

    work page 2022

  18. [18]

    Liu and Y

    Y . Liu and Y . Gao. Stability of inverse boundary value problem for the fourth-order Schr ¨odinger equation. arXiv:2512.18212, 2025

    work page arXiv 2025

  19. [19]

    W. H. McCrea and R. A. Newing. Boundary conditions for the wave equation.Proc. London Math. Soc. (2), 37:520–534, 1934

    work page 1934

  20. [20]

    H. J. Priestley. On some solution of the wave equation.Proc. London Math. Soc. (2), 20(1):37–50, 1921. INVERSE PROBLEM OF BIHARMONIC W A VE EQUATIONS 17

    work page 1921

  21. [21]

    A. G. Ramm. An inverse problem for biharmonic equation.Internat. J. Math. Math. Sci., 11(2):413–415, 1988

    work page 1988

  22. [22]

    Zhao and G

    X. Zhao and G. Yuan. Linearized inverse problem for biharmonic operators at high frequencies.Math. Methods Appl. Sci., 48(1):852–869, 2025. SCHOOL OFMATHEMATICS ANDSTATISTICS, NORTHEASTNORMALUNIVERSITY, CHANGCHUN, JILIN130024, P.R.CHINA. Email address:bimh801@nenu.edu.cn SCHOOL OFMATHEMATICS ANDSTATISTICS, CENTER FORMATHEMATICS ANDINTERDISCIPLINARYSCI- EN...

    work page 2025