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An inverse source problem for the Monge--Ampere equation from large boundary data

T0 review · 1 major / 0 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The Dirichlet-to-Neumann map determines the positive source uniquely for the Monge-Ampere equation with convex solutions.

desk verdict The paper proves uniqueness for the source in the Monge-Ampère equation from the DN map by reducing via large boundary data to X-ray injectivity; the reduction step is the part that needs verification. read the letter →

arxiv 2606.07064 v1 pith:4EMRZLJL submitted 2026-06-05 math.AP

Cătălin I. Cârstea , Tuhin Ghosh This is my paper
classification math.AP
keywords inversesourceproblemMonge-AmpereequationDirichlet-to-NeumannmapX-raytransformconvexsolutionsuniquenessboundarymeasurements
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies an inverse source problem for the equation det D²u = f(x) on a bounded smooth uniformly convex domain. It proves that the Dirichlet-to-Neumann map associated with convex solutions uniquely identifies the positive source f in the smooth classical regime. The argument proceeds by constructing a family of large boundary values. If correct, this shows that boundary observations alone suffice to recover the internal source without interior measurements.

What carries the argument

A family of large boundary values that reduces the inverse source problem for the Monge-Ampere equation to the injectivity of the Euclidean X-ray transform.

What would settle it

Two distinct positive smooth sources f and g on the same domain that produce identical Dirichlet-to-Neumann maps for every sufficiently large convex boundary datum would falsify the claim.

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Extended reading notes

Core claim

In the smooth classical regime, the Dirichlet-to-Neumann map associated with convex solutions to det D²u = f(x) determines the positive source f uniquely. The proof constructs a family of large boundary values that reduces the inverse source problem to the injectivity of the Euclidean X-ray transform.

Load-bearing premise

A family of sufficiently large boundary values can be constructed to reduce the inverse source problem to the injectivity of the X-ray transform.

Editorial extensions

If this is right

  • The positive source f is uniquely recoverable from the Dirichlet-to-Neumann map alone.
  • The uniqueness holds for convex solutions in the smooth classical regime on uniformly convex domains.
  • The reduction via large boundary data transfers known injectivity results from the X-ray transform directly to the nonlinear problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Numerical reconstruction procedures could proceed by first extracting line integrals from the large-data measurements and then inverting the X-ray transform.
  • The same large-boundary reduction strategy may extend to inverse source problems for other fully nonlinear elliptic equations.
  • The approach indicates that sufficiently strong boundary perturbations can effectively linearize the recovery of coefficients in certain nonlinear settings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper studies an inverse source problem for the Monge-Ampère equation det D²u = f(x) on a bounded smooth uniformly convex domain Ω. In the smooth classical regime, it claims to prove that the Dirichlet-to-Neumann map associated with convex solutions uniquely determines the positive source f. The argument constructs a family of large boundary values φ_t, produces corresponding convex solutions u_t, and reduces the inverse problem to the known injectivity of the Euclidean X-ray transform in a suitable scaled limit as t → ∞.

Significance. If the reduction is rigorously justified, the result would establish a uniqueness theorem for an inverse source problem in the Monge-Ampère setting by connecting it to the X-ray transform, a standard tool in integral geometry. This could be of interest in optimal transport and geometric PDEs, particularly if the construction preserves uniform convexity and controls error terms without additional assumptions on f beyond positivity and smoothness.

major comments (1)
  1. [Abstract / reduction argument] The central reduction step (abstract, final sentence) asserts that a specific family of large boundary values φ_t yields convex u_t whose DN map, after scaling, recovers the X-ray transform of log f (or an equivalent functional). However, the manuscript provides no explicit construction of φ_t, no scaling law, and no error estimates showing that boundary curvature effects and lower-order terms vanish while preserving injectivity inside the C^{2,α} convex regime. This leaves the validity of the reduction unverified and load-bearing for the uniqueness claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness in the central reduction argument. We address this point below and will incorporate the requested details in a revised version.

read point-by-point responses

  1. Referee: [Abstract / reduction argument] The central reduction step (abstract, final sentence) asserts that a specific family of large boundary values φ_t yields convex u_t whose DN map, after scaling, recovers the X-ray transform of log f (or an equivalent functional). However, the manuscript provides no explicit construction of φ_t, no scaling law, and no error estimates showing that boundary curvature effects and lower-order terms vanish while preserving injectivity inside the C^{2,α} convex regime. This leaves the validity of the reduction unverified and load-bearing for the uniqueness claim.

    Authors: We agree that the reduction requires a more self-contained and explicit presentation to allow verification of the limit process. In the revision we will add an explicit construction of the family φ_t (of the form t·ψ + lower-order correction chosen to preserve uniform convexity of the domain and the solution), state the precise scaling (normalization by t together with the associated change of variables for the DN map), and supply error estimates (via a new lemma) showing that curvature and lower-order contributions vanish in the t→∞ limit while remaining inside the C^{2,α} convex regime. These additions will make the passage to the Euclidean X-ray transform of log f fully rigorous and verifiable. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; reduction to external X-ray injectivity is independent.

full rationale

The paper reduces the inverse source problem to injectivity of the Euclidean X-ray transform via a family of large boundary values. This is an external, standard result (not derived or fitted inside the paper). No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Abstract-only; ledger entries are extracted directly from the stated assumptions and reduction step.

assumptions (2)
  • domain assumption Domain is bounded, smooth, and uniformly convex.
    Stated in the abstract as the setting for the equation.
  • domain assumption Solutions are convex (classical smooth regime).
    Required for the Dirichlet-to-Neumann map associated with convex solutions.

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Cite this review

Pith. "Pith review of An inverse source problem for the Monge--Ampere equation from large boundary data." pith.science (2026). https://pith.science/paper/4EMRZLJL

@misc{pith2026260607064,
  author       = {Pith},
  title        = {Pith review of: An inverse source problem for the Monge--Ampere equation from large boundary data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EMRZLJL}},
  note         = {Machine review of arXiv:2606.07064}
}
abstract

We study an inverse source problem for the Monge--Ampere equation \[ \det D^2u=f(x) \] on a bounded smooth uniformly convex domain. In the smooth classical regime, we prove that the Dirichlet-to-Neumann map associated with convex solutions determines the positive source uniquely. The proof uses a family of large boundary values and reduces the inverse source problem to the injectivity of the Euclidean X-ray transform.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Affine section tomography for inverse source problems in $k$-Hessian equations with restricted large boundary data

    math.AP 2026-07 accept novelty 7.0 of 10

    Restricted large-data nonlinear DN maps for k-Hessian equations recover a positive source uniquely through the affine q-plane Radon transform of its zero extension.

  2. Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation

    math.AP 2026-07 accept novelty 6.5 of 10

    If two positive curvatures share the same first boundary jet and induce the same nonlinear DN map on a common open class of admissible boundary data, then the curvatures coincide throughout a planar domain.

Reference graph

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