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On the singular set of the free boundary for a Monge-Amp\`ere obstacle problem

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Pith's one-line read Flat parts of the free boundary in a Monge-Ampère obstacle problem have Hausdorff dimension below (n+q)/2.

desk verdict New dimension bound for the singular free boundary that looks correct, but the s>1 optimality example has a broken exponent and the paper leans on an unverified companion preprint. read the letter →

arxiv 2506.08387 v1 pith:SYM5OYH3 submitted 2025-06-10 math.AP

Tianling Jin , Xushan Tu , Jingang Xiong This is my paper
classification math.AP MSC 35B2535J9635R35
keywords Monge-Ampèreequationobstacleproblemfreeboundarysingularsetdimensionestimatenon-strictconvexitystrongmaximumprinciplestabilityofcoincidencesets
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

The paper proves sharp upper bounds on the size of the flat, non-strictly convex part of the free boundary for the degenerate Monge-Ampère obstacle problem $\det D^2 v = v^q \chi_{\{v>0\}}$ with $0 \le q < n$. The main result, Theorem 1.1, says that any convex subset $E$ of the non-strictly convex free boundary satisfies $\dim E < (n+q)/2$, and that if the solution has boundary growth $v(x) \le C|x-x_0|^s$ near a boundary point $x_0 \in E$ with $s>1$, then $\dim E \le n - (n-q)s/2$. Two families of examples show both bounds are optimal: one with zero-measure coincidence set and one with positive-measure coincidence set whose singular free boundary has dimension $\lceil (n+q)/2\rceil - 1$. The paper also proves a strong maximum principle that holds near the strictly convex part of the free boundary and fails on the singular part, and it shows the coincidence set is stable under uniform convergence of solutions exactly when it has positive measure.

What carries the argument

The load-bearing mechanism is the identification of $\Gamma_{\mathrm{nsc}}$ as $\Sigma_v\cap\Gamma$: the union of all convex sets $E\subset\Omega$ on which $v$ is linear and whose extreme points lie on $\partial\Omega$, equivalently the non-trivial exposed faces of the coincidence set $K=\{v=0\}$. This structure, supplied by the companion paper [14], turns the geometric question into a volume question: Lemma 2.3 says a convex supersolution $w$ with $w\le h$ on $\partial O$ satisfies $|O|\le C(n,q)h^{(n-q)/2}$. The proof then compares the sublevel set $S_{\tilde v}^h=\{x:\tilde v(x)<h\}$ from above and below, converting the volume comparison into Hausdorff-dimension bounds. The sublevel-set lower bound uses the convex hull of $E$ and a ball of appropriate scale, plus the asserted growth factor $\omega(h)$ with $\omega(h)/h\to\infty$.

What would settle it

A convex function $v$ satisfying (4) whose non-strictly convex free boundary contains a flat face of Hausdorff dimension $\ge (n+q)/2$ would falsify Theorem 1.1, and one whose boundary growth is $|x-x_0|^s$ with a flat face of dimension $> n-(n-q)s/2$ would falsify the sharpened bound. The paper's own examples stop one integer below the extremal threshold, so the decisive calculation is whether any construction can attain exactly $(n+q)/2$. A more direct check is to compute $|S_{\tilde v}^h\cap\{x_1>0\}|$ for the given examples and verify the asserted lower bound $\omega(h)h^{n-k-1}|E|$ with $\omega(h)/h\to\infty$, since the paper does not provide the full derivation.

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Extended reading notes

Core claim

The paper's central claim is that the degeneracy exponent $q$ shifts the dimension bound for the non-strictly convex free boundary from the classical $n/2$ for the homogeneous Monge-Ampère equation to $(n+q)/2$, and that a boundary-growth condition improves the bound to $n-(n-q)s/2$. The proof uses the structural description of this singular part as a union of exposed faces of the coincidence set $K=\{v=0\}$ whose extreme points lie on $\partial\Omega$, and a volume bound for supersolutions of the form $|O|\le C h^{(n-q)/2}$. The boundary-growth condition enters because the ray from $x_0$ intersects the convex hull of $E$ at scale $h^{1/s}$, giving the smaller volume lower bound $(c h^{1/s})^{n-k}|E|$. The examples are explicit subsolutions built from functions of two radial variables, with polytope-skeleton constructions producing the positive-measure coincidence sets.

Load-bearing premise

The dimension estimate rests on the structure theory of the non-strictly convex free boundary and on the volume bound for supersolutions, both taken from the authors' companion paper without re-proof, and the proof asserts without full derivation that the sublevel set has volume at least $\omega(h)h^{n-k-1}|E|$ with $\omega(h)/h\to\infty$.

Editorial extensions

If this is right

  • For $q=0$, Theorem 1.1 recovers the classical dimension estimate $\dim E < n/2$ for singular sets of Monge-Ampère solutions, and the examples reproduce the known $q=0$ singular solutions.
  • Under the boundary growth condition with $s > 2(n-1)/(n-q)$, Remark 2.6 concludes that the non-strictly convex part of the free boundary is empty, so every free-boundary point in $\Omega$ is strictly convex.
  • The examples show that merely Lipschitz solutions exist with large flat singular sets: for $q>0$ the free boundary can be a smooth $(n-1)$-dimensional hypersurface while the solution is only Lipschitz, and for $n\ge 3$ the free boundary itself can be merely Lipschitz.
  • There exist solutions with positive-measure coincidence sets for which the singular free boundary has dimension $\lceil (n+q)/2\rceil - 1$, matching the upper bound from below.
  • Proposition 3.9 gives a stability dichotomy: the coincidence set is stable under local uniform convergence of solutions if and only if it has positive Lebesgue measure.

Reading between the lines

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

  • We infer that the exponent $(n-q)/2$ in the volume bound is the intrinsic degeneracy scale of the problem; intermediate boundary-growth rates between Lipschitz and power-type should produce dimension bounds interpolating between (5) and (7), though the paper only states the pure-power case.
  • We infer that the construction principle behind the examples is flexible: any exposed $k$-dimensional face of a convex coincidence set should be realizable as a singular free-boundary component by attaching suitable Monge-Ampère subsolutions, so the bound $\dim E < (n+q)/2$ is plausibly sharp for every integer $k < (n+q)/2$.
  • A testable extension is whether the same dimension bound holds for supersolutions rather than solutions; since the upper-bound half of the proof uses only the supersolution volume estimate, it may transfer directly, while the sharpness examples would need modification.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper studies the singular (non-strictly convex) part Γnsc of the free boundary for the Monge–Ampère obstacle problem det D^2 v = v^q χ_{v>0} with q∈[0,n). Theorem 1.1 claims that every convex subset E of Γnsc has Hausdorff dimension < (n+q)/2, with the improved bound dim E ≤ n − (n−q)s/2 when v satisfies a boundary growth condition of order s>1 at a point of E∩∂Ω. Theorems 1.2 and 1.3 give examples intended to show optimality of these bounds, by constructing solutions whose coincidence set is R^k or a polytope skeleton. Section 3 contains a strong maximum principle near the strictly convex part of the free boundary, failure of the principle at the singular set, and a stability criterion for coincidence sets under uniform convergence of solutions.

Significance. The dimension estimates are a natural and worthwhile extension of Caffarelli's classical theorem for the Monge–Ampère equation, and the examples address questions of optimality and regularity that are of current interest. The paper also identifies new phenomena (merely Lipschitz solutions, failure of W^{2,1} regularity, stability of coincidence sets) that are valuable if the proofs are correct. The central results are plausible, but the proof of the optimality examples contains a load-bearing algebraic error and a comparison-principle misapplication, and the proof of the dimension estimate relies on unproved structural inputs from the companion preprint [14]. These issues are local and appear fixable, so the manuscript warrants major revision rather than rejection.

major comments (5)
  1. [Section 2.2, proof of Theorem 1.2(ii)] The stated exponent γ = (n−k+q+(k−n+q)s)/k does not balance the determinant formula in Lemma 2.7. Direct computation gives det D^2 w ∼ ρ^{(s−2)(n−k)+γ k} (up to constants), which must be compared with w^q ∼ ρ^{q s}; the balancing condition is γ = (q s − (s−2)(n−k))/k, not the displayed formula. The difference is (n−k−q)/k. For the admissible parameters n=3, k=1, q=0, s=4/3 the printed formula gives γ = −2/3, so ρ^γ is singular at ρ=0 and the proposed function is not convex. This invalidates the optimality construction for the improved bound (7) as written.
  2. [Section 2.2, proof of Theorem 1.2 (comparison step)] In both cases (i) and (ii) the comparison principle is applied in the wrong direction. The function α w with α = c^{1/(q−n)} satisfies det D^2(α w) = α^n det D^2w ≥ α^n c w^q = α^q w^q = (α w)^q, so α w is a supersolution, not a subsolution, of the obstacle equation. Lemma 2.1 then gives α w ≥ v, the opposite of the claimed inequality c^{1/(q−n)}w ≤ v. The lower-bound growth can likely be recovered by replacing α with a sufficiently small multiple of c^{1/(q−n)}, but as written the step is invalid.
  3. [Proposition 2.9] The assertion that w = M_2 max{max_i(Φ_i+M_1ℓ_i),0} is a subsolution 'for sufficiently large M_2' is reversed. Since n>q, rescaling by M_2 gives det D^2(M_2 F) = M_2^n det D^2F while (M_2 F)^q = M_2^q F^q; the power M_2^n grows faster than M_2^q, so a small multiplier, not a large one, is needed to make the determinant inequality point in the correct direction. The proof of Theorem 1.3 relies on this subsolution construction, so the scaling needs to be corrected.
  4. [Proof of Theorem 1.1, volume lower bound] The displayed lower bound ω(h) h^{n−k−1} |E∩Ω| ≤ |S_{\tilde v}^h ∩ {x_1>0}| with ω(h)/h → ∞ is asserted without derivation. The geometric convex hull of a k-dimensional set E and a ball of radius h has n-dimensional volume of order h^{n−k}, not h^{n−k−1}; the stronger lower bound involving ω(h) appears to require additional information about the flatness of \tilde v near E that is not supplied. This bound is load-bearing for the strict inequality dim E < (n+q)/2, so a proof or a precise reference for this estimate is needed.
  5. [Section 2, Lemmas 2.3 and Proposition 2.4] The proof of Theorem 1.1 depends essentially on Lemma 2.3 (the volume bound for supersolutions) and Proposition 2.4 (the structural characterization of Γnsc as a union of exposed faces with extreme points on ∂Ω), both taken from the authors' preprint [14] without proof. Since these inputs are not verified in the present manuscript and the dimension estimate collapses if either fails, the authors should either include proofs of the required statements or clearly indicate that [14] has been accepted for publication.
minor comments (6)
  1. [Introduction, Remark 1.4] The phrasing 'C 1,α regular' should read 'C^{1,α}-regular'.
  2. [Theorem 1.2 statement] The expression 'k < n+q / 2' and the range for s are ambiguous; they should be parenthesized as k < (n+q)/2 and s ∈ [1, (2n−2k)/(n−q)].
  3. [Section 2.2, proof of Theorem 1.2(i)] The formula β = n − k + 1 + q / k + 1 is ambiguous; it should be β = (n−k+1+q)/(k+1).
  4. [Lemma 3.7] There is a typo 'begnning' in the phrase 'as proved at the begnning'; it should be 'beginning'.
  5. [Proof of Lemma 3.6] In the contradiction (12), the expression '− τ ε / (1−ε) e_n ⊂ K_1' should be '−τ ε(1−ε)^{-1} e_n ∈ K_1' with set membership rather than inclusion.
  6. [Theorem 1.2 and Remark after it] The sentence 'The estimates (5) and (6) in Theorem 1.1 are optimal' should refer to (5) and (7), since (6) is the boundary-growth assumption, not an estimate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 and the optimality examples are derived from prior lemmas, standard convex-geometric estimates, and explicit computations, not from the conclusions they purport to establish.

full rationale

I find no circular step. The central dimension estimate in Theorem 1.1 is obtained by combining Lemma 2.3 (a volume bound for supersolutions, quoted from the authors' prior work [14]) with a convex-geometric lower bound on the sublevel set S_{\tilde{v}}^h. Lemma 2.3 and Proposition 2.4 are indeed imported from [14], and the reliance on [14] is substantial, but this is heavy self-citation rather than circularity: those results have their own stated assumptions and do not contain Theorem 1.1 as an input. The proof does not redefine the target quantity in terms of itself, and no parameter is fitted to a subset of data and then renamed a prediction. The optimality constructions in Theorems 1.2 and 1.3 are explicit subsolutions converted to solutions by the comparison principle; they do not assume the dimension bound they are meant to demonstrate. I also explicitly flag a separate correctness issue that is not circularity: in the proof of Theorem 1.2(ii), the displayed identity 'det D^2w = \rho^{qs}(s+\gamma\rho^{\gamma-s}f)^{n-k-1}[s(s-1)+\gamma(\gamma-1)\rho^{\gamma-s}f-\gamma^2\rho^{\gamma-s}r^2]' is not consistent with Lemma 2.7. Direct expansion using Lemma 2.7 gives an extra factor \rho^{k+q-n}, and the stated range for s does not always ensure \gamma \ge s. Thus the optimality example for the improved bound (7) is invalid as written; this is a computational/support gap in an 'optimality' claim, not a circular reduction of the main theorem to its own inputs. Since no derivation step reduces to its own conclusion or to a fitted input, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central dimension estimate is parameter-free, but the optimality examples depend on construction constants β, γ, M1, M2. The paper inherits several key structural theorems from the authors' prior preprint [14], which are not re-proved here.

free parameters (3)
  • β = (n-k+1+q)/(k+1)
    Construction parameter in Theorem 1.2(i), chosen so that the determinant scales as ρ^q; determined by algebra, not fitted to data.
  • γ = intended: [2(n-k)+(k-n+q)s]/k (text appears to have a typo)
    Construction parameter in Theorem 1.2(ii), chosen so that the determinant scales as ρ^{qs}; the printed formula appears erroneous.
  • M1, M2 = sufficiently large/small (unspecified)
    Parameters in Proposition 2.9 used to build a subsolution from maxima of translated solutions; values are not quantified.
assumptions (5)
  • domain assumption Comparison principle (Lemma 2.1)
    Proved in [14, Lemma 2.3]; used throughout the paper to compare subsolutions and supersolutions.
  • domain assumption Volume estimate for supersolutions (Lemma 2.3)
    Proved in [14, Lemma 2.4]; critical in the proof of Theorem 1.1 and in Lemma 3.7.
  • domain assumption Structure of the singular free boundary (Proposition 2.4)
    Proved in [14, Proposition 3.1, Theorems 1.4 and 1.7]; gives that Γnsc is the union of exposed faces with extreme points on ∂Ω and provides C^{1,β} regularity away from Σ_v.
  • domain assumption Existence and uniqueness via Perron's method (Lemma 2.2)
    Proved in [14, Proposition 2.2, Corollary 2.4]; used to define solutions in the examples.
  • standard math Caffarelli's theory of strict convexity and singular sets for classical Monge-Ampère equations
    Background results from [1-4] which the paper uses as benchmarks.

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Pith. "Pith review of On the singular set of the free boundary for a Monge-Amp\`ere obstacle problem." pith.science (2026). https://pith.science/paper/SYM5OYH3

@misc{pith2026250608387,
  author       = {Pith},
  title        = {Pith review of: On the singular set of the free boundary for a Monge-Amp\`ere obstacle problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYM5OYH3}},
  note         = {Machine review of arXiv:2506.08387}
}
abstract

This is a continuation of our earlier work [14] on the Monge-Amp\`ere obstacle problem \[ \det D^2 v = v^q \chi_{\{v>0\}}, \quad v \geq 0 \text{ convex} \] with $q \in [0,n)$, where we studied the regularity of the strictly convex part of the free boundary. In this work, we examine the non-strictly convex part of the free boundary and establish optimal dimension bounds for its flat portion. Additionally, we investigate the strong maximum principle and a stability property for this Monge-Amp\`ere obstacle problem.

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