arxiv: 2605.11620 · v1 · submitted 2026-05-12 · 🧮 math.OC

Recognition: 2 theorem links

· Lean Theorem

Boundary observability for gas giant metrics

Antti Kykk\"anen, CaGE), Emmanuel Tr\'elat (LJLL (UMR\_7598), Maarten V. de Hoop

Pith reviewed 2026-05-13 01:34 UTC · model grok-4.3

classification 🧮 math.OC

keywords observability inequalitygas giant manifoldsboundary measurementsNeumann tracewave equationsperturbation argumentIngham inequalityRiemannian metrics

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The pith

An observability inequality holds for waves on gas giant manifolds from full boundary Neumann measurements.

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

This paper proves that waves on Riemannian manifolds with metrics singular at the boundary satisfy an observability inequality from Neumann-type boundary traces. Such manifolds model acoustic wave propagation inside gas giant planets, so the result supplies a mathematical basis for determining the full wave state from surface data. The argument first reduces a general gas giant metric to a separable one by perturbation while keeping observability intact. In the separable case it then combines analysis that is uniform across tangential frequencies with an Ingham inequality to obtain the bound. A reader interested in wave control or planetary interior monitoring would see why the inequality matters.

Core claim

We establish an observability inequality using full boundary measurements given by a Neumann-type trace that is natural in the gas giant setting. The proof proceeds in two steps. First, observability for a general gas giant metric is reduced to the so-called separable case via a perturbation argument. In the separable case, we employ a uniform-in-tangential-frequency analysis combined with an Ingham inequality to prove observability.

What carries the argument

Perturbation reduction of a general gas giant metric to the separable case, followed by uniform-in-tangential-frequency analysis and an Ingham inequality applied to the wave equation.

Load-bearing premise

The perturbation argument reducing a general gas giant metric to the separable case preserves the observability inequality without introducing uncontrolled error terms.

What would settle it

A concrete gas giant metric for which the observability inequality fails after the perturbation step, or a separable metric where the uniform frequency analysis plus Ingham inequality does not yield the bound.

read the original abstract

We study the observability of waves on gas giant manifolds which are a class of Riemannian manifolds whose metrics are singular at the boundary. Such manifolds arise naturally in modeling of acoustic wave propagation in gas giant planets.We establish an observability inequality using full boundary measurements given by a Neumann-type trace that is natural in the gas giant setting. The proof proceeds in two steps. First, observability for a general gas giant metric is reduced to the so-called separable case via a perturbation argument. In the separable case, we employ a uniform-in-tangential-frequency analysis combined with an Ingham inequality to prove observability.

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.

Desk Editor's Note private letter to a colleague

The paper gives an observability result for waves on singular gas giant metrics by reducing to separable case then using uniform microlocal analysis plus Ingham, but the perturbation error control in high frequencies is the part that needs checking.

read the letter

This paper proves an observability inequality for the wave equation on gas giant metrics. These are Riemannian manifolds whose metrics turn singular at the boundary and arise in acoustic modeling inside gas giant planets. The measurements come from a full-boundary Neumann-type trace that matches the setting. The proof reduces the general metric to a separable one by perturbation, then handles the separable case with microlocal analysis uniform in tangential frequencies combined with an Ingham inequality. The new element is the application to this class of singular manifolds rather than the usual smooth case. The physical motivation and the choice of trace are sensible. The separable-case argument follows standard tools adapted to the geometry and looks internally consistent. The soft spot is the perturbation step. The difference between the two wave operators has to yield an error that does not grow with frequency, otherwise the uniform constant needed for Ingham fails. The abstract gives no explicit bounds on that error in the high-frequency or semiclassical regime, so the reduction could introduce uncontrolled terms exactly where the stress-test note flags the risk. Without those estimates visible, the central claim rests on an unverified technical point. This work targets specialists in PDE control on non-smooth geometries or in geophysical wave problems. A reader already working on observability inequalities for singular manifolds would get the most from it. The claim is specific enough and the outline clear enough that it deserves a serious referee to verify the perturbation estimates and the uniformity constants rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes a boundary observability inequality for the wave equation on gas giant manifolds (Riemannian manifolds with metrics singular at the boundary, arising in acoustic modeling of gas giant planets). The inequality uses full-boundary Neumann-type trace measurements. The proof reduces the general-metric case to the separable case by a perturbation argument, then establishes the result in the separable case via microlocal analysis that is uniform in tangential frequencies combined with an Ingham inequality.

Significance. If the perturbation reduction is shown to preserve the observability constant with frequency-independent error bounds, the result would extend observability theory to a new class of singular metrics with direct geophysical modeling applications. The two-step strategy (perturbation plus uniform microlocal + Ingham) is a coherent framework, and the choice of the natural Neumann trace strengthens applicability. The paper correctly identifies the separable case as the core technical step where uniformity can be obtained.

major comments (1)

[Proof strategy and perturbation reduction section] Perturbation reduction (proof strategy paragraph and the section detailing the reduction from general to separable gas giant metric): the central claim requires that the difference between the general and separable wave operators produces an error term whose contribution to the observability inequality remains bounded independently of frequency (or of the semiclassical parameter). The abstract and strategy description give no explicit estimate controlling this remainder in the high-frequency regime; if the perturbation affects the principal symbol or boundary trace at order 1, the error can grow linearly with frequency and prevent absorption into the uniform Ingham constant. Please supply the precise norm estimate (e.g., in the energy space or via microlocal norms) showing the error is O(1) uniformly.

minor comments (2)

[Abstract] Abstract: the two-step strategy is clearly stated, but a single clause indicating that the perturbation error is controlled uniformly would help readers immediately assess the technical feasibility.
[Introduction] Notation: the precise definition of a 'gas giant metric' (including the type of singularity at the boundary) should appear in the introduction before the proof outline, rather than being deferred.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive feedback on the perturbation reduction. We address the single major comment below.

read point-by-point responses

Referee: [Proof strategy and perturbation reduction section] Perturbation reduction (proof strategy paragraph and the section detailing the reduction from general to separable gas giant metric): the central claim requires that the difference between the general and separable wave operators produces an error term whose contribution to the observability inequality remains bounded independently of frequency (or of the semiclassical parameter). The abstract and strategy description give no explicit estimate controlling this remainder in the high-frequency regime; if the perturbation affects the principal symbol or boundary trace at order 1, the error can grow linearly with frequency and prevent absorption into the uniform Ingham constant. Please supply the precise norm estimate (e.g., in the energy space or via microlocal norms) showing the error is O(1) uniformly.

Authors: We agree that an explicit frequency-independent estimate on the perturbation error is required for the argument to be complete. The manuscript outlines the reduction but does not state the precise norm bound with sufficient detail in the high-frequency regime. In the revised version we will add a new lemma in the perturbation section establishing that the difference between the general and separable operators (including the boundary trace) produces an error of order O(1) in the energy space, uniformly with respect to the semiclassical parameter. The proof of this lemma will use the fact that the perturbation is of lower order in the principal symbol together with tangential-frequency-uniform microlocal estimates already developed for the separable case; the resulting error term is then absorbed directly into the observability constant furnished by the Ingham inequality without frequency growth. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external Ingham inequality and standard perturbation

full rationale

The paper reduces the general gas giant metric to the separable case by a perturbation argument and then proves the observability inequality in the separable case using uniform-in-tangential-frequency microlocal analysis together with the external Ingham inequality. No step equates a claimed prediction or result to a fitted parameter or self-referential definition by construction. The Ingham inequality is a standard external tool, not derived within the paper or via self-citation chain. The perturbation step is presented as a standard reduction technique without evidence that error terms are absorbed by redefinition or by construction. The central claim therefore retains independent mathematical content outside its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a perturbation argument that reduces general gas-giant metrics to separable ones while preserving observability, plus the applicability of a uniform-in-frequency Ingham inequality on the separable model. No free parameters or invented entities are visible from the abstract.

axioms (2)

domain assumption Standard properties of Riemannian manifolds with singular boundary metrics admit a well-posed wave equation and Neumann trace operator.
Invoked implicitly when stating the observability inequality for gas giant manifolds.
domain assumption The Ingham inequality applies uniformly with respect to tangential frequencies on the separable model.
Used in the second step of the proof.

pith-pipeline@v0.9.0 · 5409 in / 1241 out tokens · 31522 ms · 2026-05-13T01:34:22.350422+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
The proof proceeds in two steps. First, observability for a general gas giant metric is reduced to the so-called separable case via a perturbation argument. In the separable case, we employ a uniform-in-tangential-frequency analysis combined with an Ingham inequality to prove observability.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We establish an observability inequality using full boundary measurements given by a Neumann-type trace that is natural in the gas giant setting.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

Alabau-Boussouira, P

F. Alabau-Boussouira, P. Cannarsa, and G. Leugering. Co ntrol and stabilization of degenerate wave equations. SIAM J. Control Optim. , 55(3):2052–2087, 2017

work page 2052
[2]

Baiocchi, V

C. Baiocchi, V. Komornik, and P. Loreti. Ingham-beurlin g type theorems with weakened gap condi- tions. Acta Math. Hung. , 97(1-2):55–95, 2002

work page 2002
[3]

Bardos, G

C. Bardos, G. Lebeau, and J. Rauch. Sharp suﬃcient condit ions for the observation, control, and stabilization of waves from the boundary. SIAM Journal on Control and Optimization , 30(5):1024– 1065, 1992. 25

work page 1992
[4]

E. A. Coddington and N. Levinson. Theory of ordinary diﬀe rential equations. New York, Toronto, London: McGill-Hill Book Company, Inc. XII, 429 p. (1955)., 1955

work page 1955
[5]

Colin de Verdi` ere, C

Y. Colin de Verdi` ere, C. Dietze, M. V. de Hoop, and E. Tr´ elat. Weyl formulae for some singular metrics with application to acoustic modes in gas giants. Preprint, arXiv:2406.19734 [math.AP] (2024), to appear in Annales Inst. H. Poincar´ e, 2024

work page arXiv 2024
[6]

E. B. Davies. Heat kernels and spectral theory , volume 92 of Camb. Tracts Math. Cambridge etc.: Cambridge University Press, 1989

work page 1989
[7]

M. V. de Hoop, J. Ilmavirta, A. Kykk¨ anen, and R. Mazzeo. G eometric inverse problems on gas giants. Preprint, arXiv:2403.05475 [math.DG] (2024), 2024

work page arXiv 2024
[8]

M. V. de Hoop, A. Kykk¨ anen, and E. Tr´ elat. Moving locali zed observations and Ces` aro asymptotic observability for conservative PDEs. Preprint, 2026

work page 2026
[9]

C. Dietze. The critical case for the concentration of eig enfunctions on singular Riemannian manifolds. Preprint, arXiv:2510.23520 [math.SP] (2025), 2025

work page arXiv 2025
[10]

Dietze and L

C. Dietze and L. Read. Concentration of eigenfunctions on singular Riemannian manifolds. Preprint, arXiv:2410.20563 [math.AP] (2024), 2024

work page arXiv 2024
[11]

Duprez and G

M. Duprez and G. Olive. Compact perturbations of contro lled systems. Math. Control Relat. Fields , 8(2):397–410, 2018

work page 2018
[12]

El Harraki and A

I. El Harraki and A. Boutoulout. Controllability of the wave equation via multiplicative controls. IMA J. Math. Control Inform. , 35(2):393–409, 2018

work page 2018
[13]

M. Gueye. Exact boundary controllability of 1-D parabo lic and hyperbolic degenerate equations. SIAM J. Control Optim. , 52(4):2037–2054, 2014

work page 2037
[14]

A. Haraux. S´ eries lacunaires et contrˆ ole semi-interne des vibrations d’une plaque rectangulaire. (La- cunary series and semi-internal control of vibrations of a r ectangular plate). J. Math. Pures Appl. (9), 68(4):457–465, 1989

work page 1989
[15]

A. E. Ingham. Some trigonometrical inequalities with a pplications to the theory of series. Math. Z. , 41:367–379, 1936

work page 1936
[16]

T. Kato. Perturbation theory for linear operators. Class. Math. Berlin: Springer-Verlag, reprint of the corr. print. of the 2nd ed. 1980 edition, 1995

work page 1980
[17]

Komornik and P

V. Komornik and P. Loreti. Fourier series in control theory . Springer Monogr. Math. New York, NY: Springer, 2005

work page 2005
[18]

Lasiecka and R

I. Lasiecka and R. Triggiani. Control Theory for Partial Diﬀerential Equations: Continuo us and Approximation Theories, volume I. Cambridge University Press, Cambridge, 2000

work page 2000
[19]

Lebeau and L

G. Lebeau and L. Robbiano. Exact control of the heat equa tion. Commun. Partial Diﬀer. Equations , 20(1-2):335–356, 1995

work page 1995
[20]

J.-L. Lions. Exact Controllability, Stabilization and Perturbations f or Distributed Systems . SIAM, Philadelphia, 1988

work page 1988
[21]

L. Miller. Control Theory and Partial Diﬀerential Equations , volume 1 of C.I.M.E. Summer Schools . Springer, Berlin, 2005

work page 2005
[22]

F. W. J. Olver. Asymptotics and special functions. Comp uter Science and Applied Mathematics. New York - London: Academic Press, a subsidiary of Harcourt B race Jovanovich, Publishers. XVI, 572 p. $ 39.50 (1974)., 1974. 26

work page 1974
[23]

M. Ouzahra. Controllability of the semilinear wave equ ation governed by a multipicative control. Evol. Equ. Control Theory , 8(4):669–686, 2019

work page 2019
[24]

Reed and B

M. Reed and B. Simon. Methods of modern mathematical phy sics. II: Fourier analysis, self- adjoint- ness. New York - San Francisco - London: Academic Press, a sub sidiary of Harcourt Brace Jovanovich, Publishers. XV, 361 p. $ 24.50; £ 11.75 (1975)., 1975

work page 1975
[25]

G. Teschl. Ordinary diﬀerential equations and dynamical systems , volume 140 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2012

work page 2012
[26]

Tr´ elat.Control in ﬁnite and inﬁnite dimension

E. Tr´ elat.Control in ﬁnite and inﬁnite dimension . SpringerBriefs PDEs Data Sci. Singapore: Springer, 2024

work page 2024
[27]

Tucsnak and G

M. Tucsnak and G. Weiss. Observation and control for operator semigroups . Birkh¨ auser Adv. Texts, Basler Lehrb¨ uch. Basel: Birkh¨ auser, 2009

work page 2009
[28]

A. Zettl. Sturm-Liouville theory , volume 121 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2005

work page 2005
[29]

Zhang and H

M. Zhang and H. Gao. Null controllability of some degene rate wave equations. J. Syst. Sci. Complex. , 30(5):1027–1041, 2017. 27

work page 2017