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arxiv: 2605.19142 · v1 · pith:Y55VNXC2new · submitted 2026-05-18 · 🧮 math.AP

Solutions to Monge-Amp\`ere Equations with Low Codimensional singularities

Arghya Rakshit , Aranya Sen This is my paper

Pith reviewed 2026-05-20 08:06 UTC · model grok-4.3

classification 🧮 math.AP
keywords Monge-Ampère equationsingular measureslow codimensionoptimal transportconvex potentialsweak solutionsregularity of solutions
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The pith

Solutions to Monge-Ampère equations exist with singular measures supported on low-codimension sets.

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

The paper constructs weak solutions to Monge-Ampère equations in which the associated measures include singular parts concentrated along sets of low codimension. It draws motivation from optimal transport problems whose convex potentials satisfy similar equations and then examines the regularity properties of the resulting solutions. A reader might care because these constructions address cases where mass concentrates along lower-dimensional structures, which arise naturally in geometric analysis and transport problems but fall outside classical smooth theory.

Core claim

We construct solutions to Monge-Ampère equations whose Monge-Ampère measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present examples arising from optimal transport where the potential of optimal transport maps satisfy Monge-Ampère equations similar to the ones we study.

What carries the argument

Convex potentials arising from optimal transport problems that realize prescribed singular Monge-Ampère measures in the weak sense.

If this is right

  • The Monge-Ampère measure can include singular parts supported on sets of codimension one or two while the solution remains a convex function.
  • Regularity of the solutions is limited by the presence of these singular components, with possible loss of smoothness along the support of the singular measure.
  • Examples from optimal transport supply explicit instances where the potential solves the equation with the required singular structure.

Where Pith is reading between the lines

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

  • The method may extend to constructing solutions for other fully nonlinear equations with measure data concentrated on submanifolds.
  • Numerical approximation schemes could be designed by discretizing the underlying optimal transport maps that generate the singular measures.
  • Such solutions might model physical systems in which mass or density concentrates along interfaces or lower-dimensional features.

Load-bearing premise

The constructions require the existence of convex potentials from optimal transport that produce exactly the desired singular measures while satisfying the equation weakly.

What would settle it

A concrete optimal transport problem whose resulting potential fails to satisfy the Monge-Ampère equation with the intended singular measure on a low-codimension set, or an explicit singular measure for which no convex solution exists.

Figures

Figures reproduced from arXiv: 2605.19142 by Aranya Sen, Arghya Rakshit.

Figure 1
Figure 1. Figure 1: Optimal Transport with a ‘plus’ singularity [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Displacement Interpolation between X and Y T1 : X X Q1 Ñ Y X Q1. By Remark 2.2 in [1], T ˚ 1 is continuous in the interior. This gives that there is no singular point for T ˚ except maybe along xy “ 0. Any point on x axis in Y has to get mapped to two points on X by symmetry. Thus, every point on x axis (similarly on y axis) is a singular point for T ˚. Therefore the potential of the dual optimal transport… view at source ↗
Figure 3
Figure 3. Figure 3: Displacement Interpolation between QzλQ and Q. By exploiting symmetry of the source and target domain as in example 3.1 one can see that the singular set will be a subset of the diagonals of the square λQ. The numerical model shows it is in fact a cross. Therefore the potential of the optimal transport plan satisfies an equation of the form, detpD2uq “ 1 ` fH1 pΣq. Here Σ represents the diagonals of the ta… view at source ↗
Figure 4
Figure 4. Figure 4: Displacement Interpolation between CzΓ and C. 2.2], one can see that the singular set has to be a subset of positive y axis. The numerical model supports this. Example 3.5 (Cat’s eye). Define the domains C “ tpx, yq P R 2 : |x| 2 ` |y| 2 ă 1u [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Displacement Interpolation between CzEe and C. Remark 3.6. Taking the partial Legendre transform of solutions to Monge-Ampere equation, (8) detpD2uq “ 1 ` fH1 pLq, in the x´variable. Here L is a line, like in our Cat’s eye example, satisfy an equation of the form, ∆u ˚ “ 0, expect on L and on L, Bxu ˚ has a jump discontinuity. This shows a connection between Optimal transport of non-convex domains and the … view at source ↗
Figure 6
Figure 6. Figure 6: Partial prescription of obstacle on a tetrahedron Remark 4.3. We analyze dimensions n “ 2, 3 and 4 in particular. ‚ In dimensions n “ 2, 3, our theorem above can yield one-dimensional sin￾gularities with a solution, u, solving detpD2uq “ 1 away from those singu￾larities. When comparing to [21], and [23], we can see we have a milder singular Monge-Amp`ere measure in the sense that the singularity is dis￾tri… view at source ↗
read the original abstract

We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present examples arising from optimal transport where the potential of optimal transport maps satisfy Monge-Amp\`ere equations similar to the ones we study.

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

2 major / 2 minor

Summary. The paper constructs solutions to Monge-Ampère equations whose Monge-Ampère measures contain singular components supported on low-codimensional sets (e.g., hypersurfaces or submanifolds). It studies the regularity of these solutions and motivates the constructions via examples from optimal transport, where the potentials of optimal maps satisfy similar Monge-Ampère equations in the weak (Aleksandrov) sense.

Significance. If the constructions hold, the work would extend the theory of singular solutions to fully nonlinear elliptic equations beyond the standard absolutely continuous case, providing new examples with concentrated measure parts on low-codimension supports. The optimal-transport motivation supplies concrete realizations and may connect to applications in geometric analysis and transport theory; the presence of explicit examples and regularity statements would be a strength.

major comments (2)
  1. [§3] §3 (Construction of singular solutions): The central existence claim rests on the availability of convex potentials u arising from optimal transport problems such that the Aleksandrov measure μ_u = det(D²u) dx + singular part is supported exactly on the prescribed low-codimension set. No explicit subdifferential computation or push-forward verification is supplied for the general low-codimension case; without this, it is unclear whether the absolutely continuous part remains controlled or whether convexity is preserved when the singular support is imposed.
  2. [§4] §4 (Regularity analysis): The regularity statements for the constructed solutions appear to rely on the same OT-derived potentials. If the existence step in §3 requires additional assumptions (e.g., strict convexity or specific boundary data), these must be stated explicitly, as they directly affect the applicability of the C^{1,α} or higher estimates claimed later.
minor comments (2)
  1. [Introduction] Notation for the Monge-Ampère measure should be introduced once and used consistently; the switch between det(D²u) and the Aleksandrov definition is occasionally ambiguous in the introductory paragraphs.
  2. A short table or diagram summarizing the codimension of the singular support versus the regularity obtained would improve readability of the main results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and outline the revisions we will make to improve clarity and completeness.

read point-by-point responses

  1. Referee: [§3] §3 (Construction of singular solutions): The central existence claim rests on the availability of convex potentials u arising from optimal transport problems such that the Aleksandrov measure μ_u = det(D²u) dx + singular part is supported exactly on the prescribed low-codimension set. No explicit subdifferential computation or push-forward verification is supplied for the general low-codimension case; without this, it is unclear whether the absolutely continuous part remains controlled or whether convexity is preserved when the singular support is imposed.

    Authors: We agree that additional explicit verification would strengthen the presentation. The constructions in §3 are built from convex potentials arising in optimal transport, where convexity is preserved by definition and the singular support on low-codimension sets is achieved via the prescribed target measures. Specific examples in the manuscript include subdifferential and push-forward checks, but we acknowledge that the general low-codimension case would benefit from further detail. In the revised version we will add explicit subdifferential computations and push-forward arguments to confirm that the absolutely continuous part remains controlled and the Aleksandrov measure has the claimed support. revision: yes

  2. Referee: [§4] §4 (Regularity analysis): The regularity statements for the constructed solutions appear to rely on the same OT-derived potentials. If the existence step in §3 requires additional assumptions (e.g., strict convexity or specific boundary data), these must be stated explicitly, as they directly affect the applicability of the C^{1,α} or higher estimates claimed later.

    Authors: The referee is right to note the interdependence of the sections. The potentials used satisfy the convexity and boundary conditions inherited from the underlying optimal transport problems, which are sufficient for the regularity results we invoke. To make this transparent, we will insert an explicit paragraph at the start of §4 listing the standing assumptions (including convexity type and boundary data) and confirming that they are met by the constructions of §3. This will clarify the applicability of the C^{1,α} estimates without altering the statements themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions and regularity results appear independent of self-referential inputs

full rationale

The provided abstract and context describe constructions of solutions to Monge-Ampère equations with singular measures on low-codimension supports, motivated by (but not defined via) optimal transport examples. No quoted derivation reduces a claimed result to a fitted parameter, self-citation chain, or ansatz smuggled from the authors' prior work. The central existence and regularity claims are presented as constructions rather than predictions forced by the inputs. The OT motivation is external and does not create a self-definitional loop or load-bearing self-citation within the paper's own equations. This is the expected honest non-finding for a construction-focused manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard background results from the theory of Monge-Ampère equations and optimal transport without introducing new free parameters or invented entities.

axioms (1)
  • domain assumption Existence of convex potentials in optimal transport that satisfy a Monge-Ampère equation in the weak sense.
    The motivation section relies on this standard fact from optimal transport theory.

pith-pipeline@v0.9.0 · 5574 in / 1286 out tokens · 70608 ms · 2026-05-20T08:06:51.554252+00:00 · methodology

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