{"id":"a9deaf01-e0b8-45d0-9b8d-40de7a384886","arxiv_id":"2506.08387","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For the Monge-Ampère obstacle problem, any flat piece of the non-strictly convex free boundary has dimension less than (n+q)/2, and this bound is optimal.","lead":"This paper studies the singular part of the free boundary in a nonlinear obstacle problem governed by the Monge-Ampère equation. It proves sharp bounds on how large flat pieces of that singular boundary can be, and it shows the bounds cannot be improved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2(ii) is internally inconsistent: the stated exponent gamma does not balance Lemma 2.7, so the optimality construction for the improved bound (7) collapses as written.","rationale":"The reader's weakest_assumption concerns the proof of Theorem 1.1 and reliance on the authors' preprint [14]; those are legitimate gaps. However, the most decisive, checkable defect is in the optimality construction for Theorem 1.2(ii): the stated γ is algebraically inconsistent with Lemma 2.7, and for n=3, k=1, q=0, s=4/3 it gives a negative exponent, so the alleged subsolution is not even finite on the coincidence set. Since Theorems 1.2 and 1.3 are the only evidence for the optimality assertions of Theorem 1.1, the central optimality claim is not supported as written. A second, independent defect in the same theorem is that the comparison chain v ≥ A w with A=c^{1/(q-n)} is reversed: Lemma 2.1 gives v ≤ A w when A w is a supersolution, so the lower bound c dist^s ≤ v is also not derived. These are internal inconsistencies rather than disagreements with prior literature. The γ error appears repairable by rebalancing exponents, so a conditional major revision is appropriate rather than outright rejection; if the corrected construction fails, the optimality claims would need to be withdrawn or restricted. The reader's rationale did flag a scaling error in the subsolution construction, so there is partial agreement, but the reader's stated weakest assumption pointed elsewhere.","tokens_in":11866,"tokens_out":31402,"duration_ms":375487,"concrete_test":"Fix n=3, k=1, q=0, s=4/3 and compute det D^2w directly from Lemma 2.7 for u(ρ,r)=ρ^s+ρ^γ(1+r^2/2) with the paper's γ=2-2s. Since γ=-2/3, the ansatz blows up on R^k and fails the hypotheses; then substitute the exponent balance γ=4/3 required by Lemma 2.7 and check whether the corrected subsolution satisfies det D^2w ≥ c w^q and, after rescaling and comparison, yields the claimed two-sided bound cρ^s ≤ v ≤ Cρ^s. If the corrected construction fails, Theorem 1.2(ii) is not salvageable.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that the dimension estimates in Theorem 1.1 are optimal depends on Theorem 1.2. Its second case (s>1) proposes u(ρ,r)=ρ^s+ρ^γ f(r) with f(r)=1+r^2/2 and γ=(n-k+q+(k-n+q)s)/k. Applying Lemma 2.7 to this ansatz gives det D^2w = ρ^{(s-2)(n-k)+γk}(s+γρ^{γ-s}f)^{n-k-1}[s(s-1)+γ(γ-1)ρ^{γ-s}f-γ^2ρ^{γ-s}r^2], where the first ρ-exponent comes from the factors (u_ρ/ρ)^{n-k-1}(u_r/r)^{k-1} and the cross term. Since w^q ∼ ρ^{qs}, balancing exponents requires γ = [qs-(s-2)(n-k)]/k, not the paper's formula; the difference is (n-k-q)/k. For the admissible choice n=3, k=1, q=0, s=4/3 (allowed because s ≤ (2n-2k)/(n-q)=4/3), the paper's formula gives γ=2-2s=-2/3<0, so ρ^γ diverges at ρ=0 and the proposed function is neither finite on the coincidence set R^k nor convex. Thus the construction intended to demonstrate optimality of estimate (7) is invalid as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12156,"tokens_out":20071,"duration_ms":230816,"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":[{"comment":"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.","section":"Section 2.2, proof of Theorem 1.2(ii)"},{"comment":"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.","section":"Section 2.2, proof of Theorem 1.2 (comparison step)"},{"comment":"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.","section":"Proposition 2.9"},{"comment":"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.","section":"Proof of Theorem 1.1, volume lower bound"},{"comment":"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.","section":"Section 2, Lemmas 2.3 and Proposition 2.4"}],"minor_comments":[{"comment":"The phrasing 'C 1,α regular' should read 'C^{1,α}-regular'.","section":"Introduction, Remark 1.4"},{"comment":"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)].","section":"Theorem 1.2 statement"},{"comment":"The formula β = n − k + 1 + q / k + 1 is ambiguous; it should be β = (n−k+1+q)/(k+1).","section":"Section 2.2, proof of Theorem 1.2(i)"},{"comment":"There is a typo 'begnning' in the phrase 'as proved at the begnning'; it should be 'beginning'.","section":"Lemma 3.7"},{"comment":"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.","section":"Proof of Lemma 3.6"},{"comment":"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.","section":"Theorem 1.2 and Remark after it"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is heavily self-referential: Theorem 1.1 depends on Proposition 2.4 and Lemma 2.3 of the companion preprint [14]. If [14] is not yet accepted, this is a significant risk; the editor may wish to verify its status. The algebraic error in Theorem 1.2(ii) and the reversed comparison inequality are serious but appear to be local and repairable; the scaling issue in Proposition 2.9 is also a straightforward fix. None of the identified problems seems to require abandoning the main approach."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: the main theorem (the dimension bound for flat parts of the singular free boundary) looks new and plausible, but the optimality construction for the s>1 case has a real bug, and the paper leans heavily on the authors' own unpublished preprint [14] for load-bearing structural results. I would not reject the paper; I would send it to review and ask for repairs.\n\nWhat's genuinely new: Theorem 1.1 extends Caffarelli's estimate dim E < n/2 for the classical Monge-Ampere equation to the obstacle problem det D^2v = v^q chi_{v>0}, giving dim E < (n+q)/2, with an improved bound if the boundary data controls the solution near a point of E. The proof is a natural reduction to a supersolution and uses the volume estimate from [14]. The examples for s=1 (cylinders with linear growth) and the polyhedral construction in Theorem 1.3/Prop 2.9 are interesting and extend known constructions (Caffarelli-Yuan, Mooney-Rakshit). Section 3 on the strong maximum principle and stability is a useful addition.\n\nThe soft spots are real. In Theorem 1.2(ii), the exponent gamma is wrong. Applying Lemma 2.7 to u = rho^s + rho^gamma f(r) gives det ~ rho^{(s-2)(n-k)+gamma k} times a bounded factor. To match w^q ~ rho^{qs}, you need gamma = [qs - (s-2)(n-k)]/k, not the formula in the paper; the difference is (n-k-q)/k. For the allowed parameters n=3, k=1, q=0, s=4/3, the paper's gamma = -2/3, so rho^gamma blows up at rho=0 and the subsolution is not finite on K. So the proof of optimality of (7) collapses. The s=1 case and Theorem 1.3 don't use this construction, so they may survive.\n\nSecond, the proof of Theorem 1.1 asserts a volume lower bound of the form omega(h) h^{n-k-1} |E| with omega(h)/h -> infinity, but doesn't derive it. The geometric intuition (a tube around the k-plane) suggests h^{n-k}, not h^{n-k-1}; the omega(h) factor is doing work and needs justification.\n\nThird, the paper takes the comparison principle, volume bound, and the entire structure theory of Gamma_nsc from [14] without re-proof. That's a heavy self-citation in a load-bearing role. A referee should ask for verification of [14] or a summary of the relevant arguments.\n\nOn the reader's 'reversed inequality' concern: I checked that comparison step; the direction is fine. The real problem is the gamma exponent.\n\nWho's this for? PDE/free-boundary people, especially those working on Monge-Ampere obstacle problems and singular sets. It deserves a serious referee; the main theorem is important if it holds. Recommendation: send to review, but require a corrected optimality section and a clearer accounting of [14].","headline":"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.","tokens_in":12682,"tokens_out":14974,"would_cite":false,"duration_ms":165867,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B25","35J96","35R35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Flat parts of the free boundary in a Monge-Ampère obstacle problem have Hausdorff dimension below (n+q)/2.","keywords":["Monge-Ampère equation","obstacle problem","free boundary","singular set","dimension estimate","non-strict convexity","strong maximum principle","stability of coincidence sets"],"falsifier":"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.","tokens_in":11655,"feed_emoji":"📐","tokens_out":13030,"duration_ms":117458,"temperature":0.7,"pith_summary":"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.","feed_headline":"Flat free-boundary patches stay below (n+q)/2 dimensions","feed_subtitle":"Optimal examples show the cap is sharp, so the nonsmooth part of the contact set can be no larger.","key_machinery":"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 $\\omega(h)/h\\to\\infty$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the structural description of $\\Gamma_{\\mathrm{nsc}}$ as a union of exposed faces and the volume bound Lemma 2.3 used in the proof.","marker":"[14]"},{"why":"It is the classical dimension estimate for singular sets of Monge-Ampère solutions that Theorem 1.1 extends to the obstacle problem.","marker":"[4]"},{"why":"It provides the $q=0$ singular solutions whose explicit subsolution construction is adapted in Theorem 1.2.","marker":"[5]"},{"why":"It establishes the regularity theory for the classical Monge-Ampère obstacle problem that motivates the free-boundary questions here.","marker":"[19]"},{"why":"It supplies the strong maximum principle for Monge-Ampère equations used in Propositions 3.4 and 3.5.","marker":"[13]"}],"fun_headline_variants":["Flat free-boundary parts stay within (n+q)/2","Boundary growth sharpens free-boundary dimension cap","Sharp dimension bound for non-strict free boundary","Monge-Ampère free boundary: flat pieces capped sharply"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $\\omega(h)/h\\to\\infty$.","fun_headline_variants_meta":{"raw":{"variants":["Flat free-boundary parts stay within (n+q)/2","Boundary growth sharpens free-boundary dimension cap","Sharp dimension bound for non-strict free boundary","Monge-Ampère free boundary: flat pieces capped sharply"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1518,"prompt_tokens":859,"completion_tokens":659,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":593}},"tokens_in":475,"tokens_out":659,"duration_ms":7425,"temperature":1.0,"reasoning_tokens":593,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:14:56.274151+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Regularity and classification of the free boundary for a Monge-Amp\\`ere obstacle problem","cited_arxiv_id":"2504.21253","evidence_quote":"It supplies the structural description of $\\Gamma_{\\mathrm{nsc}}$ as a union of exposed faces and the volume bound Lemma 2.3 used in the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the classical dimension estimate for singular sets of Monge-Ampère solutions that Theorem 1.1 extends to the obstacle problem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the $q=0$ singular solutions whose explicit subsolution construction is adapted in Theorem 1.2."},{"cited_title":"Savin, The obstacle problem for Monge Ampere equation, Calc","cited_arxiv_id":null,"evidence_quote":"It establishes the regularity theory for the classical Monge-Ampère obstacle problem that motivates the free-boundary questions here."},{"cited_title":"Jian and X","cited_arxiv_id":null,"evidence_quote":"It supplies the strong maximum principle for Monge-Ampère equations used in Propositions 3.4 and 3.5."}],"review_version":1}