{"id":"0844a631-8c56-4d16-a369-10d078d9eb4e","arxiv_id":"1908.06396","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For det D^2u = F(x,u) with u=0 on a bounded convex domain, the authors prove global Hölder continuity with exponent (β-n+1)/(n+α), improved by the (a,η) convexity parameter, without requiring smoothness of the domain or F.","lead":"This paper proves existence, uniqueness, and global Hölder continuity for solutions of a degenerate and singular Monge-Ampère equation on arbitrary bounded convex domains, with only a continuous right-hand side. It gives explicit formulas linking the Hölder exponent to the convexity of the domain, extending earlier results that required smooth strictly convex domains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 1 of Theorem 1.1 applies Cheng–Yau to subdomains Ω_i using an upper bound assumed only with dist(·,∂Ω), not dist(·,∂Ω_i); for β>n+1 the two bounds are not comparable, so the uniform estimates (2.11) and the existence chain are unsupported.","rationale":"The reader's weakest assumption matches the same step. I see no stronger objection: the barrier computations in Lemmas 2.2 and the proofs of Theorems 1.2–1.3 are explicit and checkable; the smoothing step in Step 2 is standard though not fully detailed; the Caffarelli bootstrap in Step 3 is standard once F(x,u(x)) is Hölder and positive. The only place where a false or unjustified hypothesis would break the chain is the application of Cheng–Yau and of Lemma 2.2 to subdomains whose boundary distance is not the one appearing in (1.3). This is a genuine proof gap but not a demonstrated falsehood of the theorem; CONDITIONAL remains the right verdict.","tokens_in":13697,"tokens_out":18487,"duration_ms":199660,"concrete_test":"Fix n=2, Ω=B_1(0), Ω_i=B_{1-1/i}(0), β>n+1, and F(x,t)=A d_Ω(x)^{β-n-1}|t|^{-α}. For x∈Ω_i with d_{Ω_i}(x)=ε, d_Ω(x)≈1/i, so F(x,t)/(d_{Ω_i}(x)^{β-n-1}|t|^{-α}) → ∞ as ε→0; this shows Cheng–Yau's hypothesis with d_{∂Ω_i} is not inherited from (1.3). To settle whether the theorem still holds, replace the Cheng–Yau invocation with a direct Perron existence proof on Ω_i and check whether a uniform boundary Hölder estimate can be derived from the original d_Ω-bound; if the estimate fails for this example or cannot be derived, Step 1's conclusion is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is the transition from smooth strictly convex approximating domains to a general bounded convex Ω in Step 1 of the proof of Theorem 1.1. Cheng–Yau's theorem is invoked on each Ω_i, but that theorem requires (1.3) with distance to ∂Ω_i. Assumption (1.3) is made only with distance to ∂Ω. Because Ω_i⊂Ω, for x near ∂Ω_i one has d_{Ω_i}(x)≪d_Ω(x); for β>n+1 the power d^{β-n-1} is increasing, so F may fail the required bound with d_{Ω_i}. In particular, Lemma 2.2, which constructs barriers using the domain's own distance function, cannot be applied to u_i on Ω_i with the original F; the uniform estimate (2.11) is therefore not established. Since the convergence argument for u_i and the uniqueness step both rely on these uniform bounds, the central existence/regularity theorem for non-smooth domains is incomplete as written. The gap is local and plausibly repairable—for instance by a direct Perron construction or by reworking the barrier to use d_Ω—but it is exactly the step that removes the smoothness assumption on Ω.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the Dirichlet problem det D^2u = F(x,u) in a bounded convex domain Ω with u=0 on ∂Ω, where F is positive, continuous, non-decreasing in u, and satisfies the boundary-degeneracy bound (1.3). The main results are existence, uniqueness, and global Hölder continuity of the convex Alexandrov solution. Theorem 1.1 gives the exponent γ1 = (β−n+1)/(n+α) for an arbitrary bounded convex domain, with the case β ≥ α+2n−1 allowing any exponent in (0,1). Theorem 1.2 refines the exponent to γ2 = γ1 + (2n−2)/(a(n+α)) for (a,η)-type domains with a>2, and Theorem 1.3 gives γ3 = β/(n+α) under an exterior sphere condition together with a boundary lower bound under an interior sphere condition. The proofs use explicit barrier functions, the comparison principle for Alexandrov solutions, and a limiting argument from smooth strictly convex subdomains.","tokens_in":13896,"tokens_out":19817,"duration_ms":180566,"significance":"If the theorem is established, the paper gives a clean extension of Cheng-Yau's classical result by removing the strict convexity, smoothness, and differentiability assumptions, and it provides a quantitative relation between the boundary Hölder exponent and the convexity of the domain. The barrier constructions are explicit, the exponent algebra is internally consistent, and the results are falsifiable. The proofs are largely self-contained, and the reliance on the authors' earlier work [11] is mitigated by reproducing Lemma 2.1. The main strength is the systematic derivation of boundary decay from simple geometric barriers, which is a genuine contribution if the gap in the approximation argument is repaired.","major_comments":[{"comment":"The existence and uniform-estimate chain for non-smooth domains is not justified as written. Cheng-Yau's theorem is invoked on each strictly convex C^2 subdomain Ω_i, but that theorem requires the upper bound (1.3) with distance to ∂Ω_i, while assumption (1.3) is stated only with d_x = dist(x,∂Ω). For β−n−1 > 0, the two bounds are not comparable: since Ω_i ⊂ Ω we have d_{Ω_i}(x) ≤ d_Ω(x), and F ≤ A d_Ω^{β−n−1}|t|^{-α} does not imply F ≤ A d_{Ω_i}^{β−n−1}|t|^{-α}. The subsequent application of Lemma 2.2 to u_i on Ω_i has the same defect: the proof of Lemma 2.2 depends on the inequality d_x ≤ x_n after normalizing at a boundary point of the domain in question, and when one normalizes at z_i ∈ ∂Ω_i the global distance d_Ω is not controlled by x_n because z_i is an interior point of Ω. Consequently the uniform bound (2.11) is not established, and the convergence argument that constructs the solution on Ω is incomplete. This step is exactly what removes the smoothness assumption on Ω. The issue is local and plausibly repairable, for example by constructing the barrier directly from ∂Ω, but it must be fixed before Theorem 1.1 is established.","section":"Section 2, Step 1"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this paper has a genuine new result and a genuine missing argument. The new part is the quantitative link between boundary geometry and the Hölder exponent: Theorem 1.2 gives γ = (β−n+1+(2n−2)/a)/(n+α) for (a,η)-type domains with a>2, and Theorem 1.3 recovers the exterior-sphere case by letting a=2. That is a real extension of Cheng–Yau [4], which needed C^2 strictly convex domains and smooth F. The barrier calculations are explicit, the exponents are derived rather than fitted, and the comparison arguments are standard. Anyone working on degenerate singular Monge–Ampère equations will want to know these statements.\n\nThe soft spot is Step 1 of Theorem 1.1. The proof exhausts Ω by smooth strictly convex subdomains Ω_i⊂Ω, applies Cheng–Yau to get u_i on each Ω_i, and then invokes Lemma 2.2 for the uniform estimate. Lemma 2.2 is stated for a domain whose F satisfies (1.3) with its own distance function. The assumption gives (1.3) with d_Ω, not d_{Ω_i}. When β>n+1, d_{Ω_i}^{β−n−1} can be much smaller than d_Ω^{β−n−1} near corners; the transfer fails. This is a real gap in the written proof, and it is exactly the step that removes smoothness of Ω.\n\nI think it is repairable. The barrier in Lemma 2.2 can be run with the supporting hyperplane from ∂Ω rather than ∂Ω_i; then W is a subsolution on each Ω_i and the estimate comes out as |u_i|≤C d_Ω^γ. Or one can exhaust from outside, where d_{Ω_i}≥d_Ω and (1.3) is preserved. The paper does neither, so a referee should ask for this rewrite. Secondary items are smaller: the mollification of F in Step 2 needs an explicit extension argument, and the C^{2,γ} bootstrap in Step 3 is sketched but standard.\n\nNet: the central claims are very likely correct, but the current text is not. This paper deserves a serious referee, not a desk rejection. I would send it out with a request to repair Step 1 and to state the distance comparison explicitly. If fixed, I would cite it; as posted, I would not rely on it.","headline":"A genuine extension of Cheng–Yau with explicit convexity-dependent Hölder exponents, but the proof has a repairable gap in the approximation step.","tokens_in":14506,"tokens_out":16942,"would_cite":false,"duration_ms":177193,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35J96","53A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves existence, uniqueness, and global Hölder continuity for a degenerate/singular Monge–Ampère Dirichlet problem on arbitrary bounded convex domains.","keywords":["Monge-Ampère equation","degenerate and singular boundary","global Hölder regularity","convex domain","(a,η)-type domain","exterior sphere condition","comparison principle","Alexandrov solution"],"falsifier":"Take a bounded convex domain with a flat side (e.g. an isosceles triangle in $\\mathbb{R}^2$) and $F=d_x^{\\beta-3}|u|^{-\\alpha}$ with $\\beta>n+1$. If the solution's boundary Hölder exponent is strictly smaller than $\\gamma_1=(\\beta-n+1)/(n+\\alpha)$, or if the approximating solutions on smooth subdomains fail to satisfy a uniform $C^{\\gamma_1}$ bound, then the central claim is false.","tokens_in":13430,"feed_emoji":"📐","tokens_out":6565,"duration_ms":56016,"temperature":0.7,"pith_summary":"The paper treats the Dirichlet problem $\\det D^2u=F(x,u)$ in a bounded convex domain $\\Omega$ with $u=0$ on $\\partial\\Omega$, where the right-hand side may degenerate or blow up near the boundary: $02$, the exponent improves to $\\gamma_2=\\gamma_1+(2n-2)/(a(n+\\alpha))$; if an exterior sphere condition holds, it becomes $\\gamma_3=\\beta/(n+\\alpha)$. Thus less flat boundary points yield higher Hölder exponents. The proof obtains these exponents by constructing explicit subsolutions, so it works for merely continuous $F$ and for non-smooth convex domains.","pith_inferences":["A natural next question is whether the optimal Hölder exponent on a given domain is determined by its flattest boundary point in the $(a,\\eta)$-sense; the formulas here suggest such an interpolation but the paper does not address it.","The same explicit-barrier strategy might transfer to other fully nonlinear equations whose right-hand side has a comparable power-law singularity in the distance function, such as Hessian equations with similar boundary weights.","A numerical check on a domain with a flat side could test whether the predicted exponent is sharp or whether the true solution is smoother than the barrier bound."],"forward_implications":["If correct, Theorem 1.1 extends the classical existence and Hölder regularity result from smooth strictly convex domains to arbitrary bounded convex domains and merely continuous right-hand sides.","The Hölder exponent is explicit in $\\alpha$, $\\beta$, $n$, and the convexity parameter $a$, so it can be read off from the data without knowing the solution.","On exterior-sphere domains the upper bound $\\beta/(n+\\alpha)$ is matched by a lower bound of the same power under an interior sphere condition, so the exponent is optimal in that geometric setting.","When $F$ is Lipschitz, the interior $C^{2,\\gamma}$ conclusion follows from the boundary estimate by known interior regularity, so the boundary control is the essential new step."],"supporting_citations":[{"why":"It supplies existence of convex generalized solutions on smooth strictly convex $C^2$ domains, the starting point of the exhaustion argument.","marker":"[4]"},{"why":"It introduces the $(a,\\eta)$-type convexity condition and the convexity lemma that turns boundary-distance bounds into Hölder continuity.","marker":"[11]"},{"why":"It provides the comparison principle and convergence results for generalized solutions used in the limit passages.","marker":"[9]"},{"why":"It gives the interior $C^{2,\\alpha}$ regularity used to upgrade to $C^{2,\\gamma}$ when $F$ is Lipschitz.","marker":"[1]"},{"why":"It supplies the Alexandrov solution framework and comparison principle for the boundary value problem.","marker":"[8]"}],"fun_headline_variants":["Less flat boundaries yield smoother Monge-Ampère solutions","Explicit Hölder exponents tie Monge-Ampère regularity to boundary shape","Boundary curvature elevates Hölder exponents for Monge-Ampère","Monge-Ampère solutions: Hölder exponents set by boundary flatness","New explicit exponents for degenerate Monge-Ampère Hölder regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The existence step relies on applying a classical existence theorem on each smooth approximating subdomain, but that theorem assumes the growth bound on $F$ is measured by distance to the subdomain's boundary, whereas the paper only assumes it with distance to the original boundary; for $\\beta>n+1$ the two distances give different bounds, and the transfer is not justified in the text.","fun_headline_variants_meta":{"raw":{"variants":["Less flat boundaries yield smoother Monge-Ampère solutions","Explicit Hölder exponents tie Monge-Ampère regularity to boundary shape","Boundary curvature elevates Hölder exponents for Monge-Ampère","Monge-Ampère solutions: Hölder exponents set by boundary flatness","New explicit exponents for degenerate Monge-Ampère Hölder regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000705,"raw_usage":{"total_tokens":3089,"prompt_tokens":768,"completion_tokens":2321,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":384,"completion_tokens_details":{"reasoning_tokens":2227}},"tokens_in":384,"tokens_out":2321,"duration_ms":19353,"temperature":1.0,"reasoning_tokens":2227,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:49:39.595329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a bounded convex domain with a flat side (e.g. an isosceles triangle in $\\mathbb{R}^2$) and $F=d_x^{\\beta-3}|u|^{-\\alpha}$ with $\\beta>n+1$. If the solution's boundary Hölder exponent is strictly smaller than $\\gamma_1=(\\beta-n+1)/(n+\\alpha)$, or if the approximating solutions on smooth subdomains fail to satisfy a uniform $C^{\\gamma_1}$ bound, then the central claim is false.","supporting_citations":[{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"It supplies existence of convex generalized solutions on smooth strictly convex $C^2$ domains, the starting point of the exhaustion argument."},{"cited_title":"Differential Equations, 264 (2018), 6873-6890","cited_arxiv_id":null,"evidence_quote":"It introduces the $(a,\\eta)$-type convexity condition and the convexity lemma that turns boundary-distance bounds into Hölder continuity."},{"cited_title":"E., The Monge-Amp` ere equation, Birkhauser, Boston, 2 001","cited_arxiv_id":null,"evidence_quote":"It provides the comparison principle and convergence results for generalized solutions used in the limit passages."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the interior $C^{2,\\alpha}$ regularity used to upgrade to $C^{2,\\gamma}$ when $F$ is Lipschitz."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the Alexandrov solution framework and comparison principle for the boundary value problem."}],"review_version":1}