On A Class of Degenerate And Singular Monge-Amp\`ere Equations

Huaiyu Jian; Xushan Tu; You Li

On A Class of Degenerate And Singular Monge-Amp\`ere Equations

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Pith's one-line read This paper proves existence, uniqueness, and global Hölder continuity for a degenerate/singular Monge–Ampère Dirichlet problem on arbitrary bounded convex domains.

desk verdict A genuine extension of Cheng–Yau with explicit convexity-dependent Hölder exponents, but the proof has a repairable gap in the approximation step. read the letter →

arxiv 1908.06396 v1 pith:7EDVAX5W submitted 2019-08-18 math.AP

Huaiyu Jian , You Li , Xushan Tu This is my paper · ORCID

classification math.AP MSC 35J6035J9653A15

keywords Monge-AmpèreequationdegenerateandsingularboundaryglobalHölderregularityconvexdomain(aη)-typeexteriorsphereconditioncomparisonprincipleAlexandrovsolution

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 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: $0

What carries the argument

The load-bearing object is an explicit barrier function adapted to the local shape of the domain. After placing a boundary point at the origin with $\Omega\subseteq\{x_n\ge\eta|x'|^a\}$, the paper sets $W(x)=-\big((x_n/\varepsilon)^{2/a}-|x'|^2\big)^{1/b}$ and computes its Monge–Ampère determinant in closed form. Choosing $b$ so that the powers of $x_n$ cancel makes $\det D^2W\ge F(x,W)$, so $W$ is a subsolution; the comparison principle bounds the solution by $|W|$, i.e. by a power of the distance to the boundary. A one-dimensional convexity lemma converts such boundary-distance bounds into a global Hölder estimate. Existence is obtained by exhausting the domain by smooth strictly convex subdomains, solving there, and passing to the limit using uniform Hölder bounds.

What would settle it

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.

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Extended reading notes

Core claim

The central discovery is that the global Hölder exponent is governed by the balance between the vanishing or blow-up of $F$ and the flatness of the boundary. For any bounded convex domain the solution belongs to $C^{\gamma_1}(\overline{\Omega})$ with $\gamma_1=(\beta-n+1)/(n+\alpha)$ in the main range $\beta<\alpha+2n-1$. If the boundary is of $(a,\eta)$-type with $a>2$, 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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

I 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.

Net: 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.

Referee Report

1 major / 0 minor

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.

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 (1)

[Section 2, Step 1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Hölder exponents are derived from explicit barrier constructions; self-citations to [11] are non-load-bearing.

full rationale

The derivation chain is self-contained. Theorem 1.1 is proved by constructing explicit sub-solutions W = -M x_n^γ sqrt(N^2 l^2 - r^2) in Lemma 2.2, choosing γ = (β - n + 1)/(n + α) from the algebraic condition (n + α)γ - (β - n + 1) = 0 following Eq. (2.9), then applying the comparison principle and the reproduced Lemma 2.1. Lemma 2.1 is attributed to [11] but its proof is copied in the text, so the citation is not load-bearing. The (a,η) condition is Definition 1.1 and functions as a geometric hypothesis, not as an input derived from the conclusion. Theorem 1.2 constructs W = -[(x_n/ε)^{2/a} - r^2]^{1/b} and chooses b = 2(n + α)/(a(β - n + 1) + 2n - 2) via Eq. (3.10), and the Hölder exponent γ2 is read off from the barrier; Theorem 1.3 uses a standard exterior-sphere barrier. The only reliance on the authors' prior work [11] is for the definition of (a,η) type, the reproduced Lemma 2.1, and the parenthetical equivalence between (2,η) domains and the exterior sphere condition; none is the central claim, and each is replaceable by displayed arguments. A correctness gap exists in Step 1 of Theorem 1.1: Cheng-Yau's theorem is applied to approximating domains Ω_i while assumption (1.3) is stated with d_Ω, and for β > n + 1 the two distance functions are not comparable. This is a gap in justification, not a circular reduction; no equation is reintroduced as its own input. Therefore the paper's results are not circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard PDE machinery: comparison principles, stability, and interior regularity for Monge-Ampère. The only nonstandard input is the (a,η)-type convexity notion from the authors' prior paper [11], which is used as a definition rather than an unproved fact. No free parameters are fitted.

assumptions (4)

standard math Comparison principle for convex generalized (Alexandrov) solutions of Monge-Ampère equations with monotone right-hand side
Used throughout to compare the explicit barrier sub-solutions with the solution; cited to [8,9,23] in Section 2 and in the proofs of Theorems 1.1-1.3.
standard math Cheng-Yau existence theorem (Theorem 5 in [4]) for strictly convex C^2 domains and smooth F satisfying (1.2)-(1.3)
Provides the existence of approximating solutions u_i in Step 1 of Theorem 1.1; also the baseline existence for smooth data.
standard math Stability of convex generalized solutions under uniform convergence (Lemma 1.6.1 in [9])
Used to pass from approximating solutions u_i and u_j to the limit solution u in Steps 1 and 2 of Theorem 1.1.
standard math Caffarelli's interior C^{2,α} regularity for Monge-Ampère equations with positive Hölder right-hand side
Used in Step 3 of Theorem 1.1 to bootstrap to interior C^{2,γ1}; the cited [1] is actually the W^{2,p} paper, with [15] as additional proof.

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Pith. "Pith review of On A Class of Degenerate And Singular Monge-Amp\`ere Equations." pith.science (2026). https://pith.science/paper/7EDVAX5W

@misc{pith2026190806396,
  author       = {Pith},
  title        = {Pith review of: On A Class of Degenerate And Singular Monge-Amp\`ere Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7EDVAX5W}},
  note         = {Machine review of arXiv:1908.06396}
}

abstract

In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Amp\`ere type equations which may be degenerate and singular on the boundary of convex domains. We will establish a relation of the H$\ddot{o}$lder exponent for the solutions with the convexity for the domains.

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Works this paper leans on

24 extracted references · 24 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

Pure Appl

Cheng, S.Y., Yau, S.T., On the regularity of the Monge-Amp` ere eq uation det ∂ 2u ∂x i∂x j = F (x, u), Comm. Pure Appl. Math. 30 (1977), 41–68

work page 1977

[2] [11]

Diﬀerential Equations, 264 (2018), 6873-6890

Jian, H.Y., Li, Y., Optimal boundary regularity for a Singular Monge -Amp` ere equation, J. Diﬀerential Equations, 264 (2018), 6873-6890

work page 2018

[3] [1]

Caﬀarelli, L.A., Interior W 2,p estimates for solutions of Monge-Amp` ere equations, Ann. Math. 131 (1990), 135-150

work page 1990

[4] [2]

Pure Appl

Caﬀarelli, L.A., Nirenberg, L., Spruck, J., The Dirichlet problem for n onlinear second-order elliptic equations I, Monge-Amp` ere equation, Comm. Pure Appl. Mat h. 37 (1984), 369–402

work page 1984

[5] [3]

Calabi, E., Complete aﬃne hypersurfaces I, Symposia Mathematic a 10 (1972), 19-38

work page 1972

[6] [5]

Pure Ap pl

Cheng, S.Y., Yau, S.T., On the existence of a complete K¨ ahler metr ic on noncompact complex manifolds and the regularity of Feﬀerman’s equation, Comm. Pure Ap pl. Math. 33 (1980), 507–544

work page 1980

[7] [6]

Pure Appl

Cheng, S.Y., Yau, S.T., Complete aﬃne hypersurfaces I, The comp leteness of aﬃne metrics, Comm. Pure Appl. Math. 39 (1986), 839-866

work page 1986

[8] [7]

Chou, K.S., Wang, X.-J., The Lp-Minkowski problem and the Minkowski problem in cen- troaﬃne geometry, Adv. Math. 205 (2006), 33–83

work page 2006