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REVIEW 3 major objections 6 minor 45 references

Removing singularities for fully nonlinear PDEs

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A viscosity solution whose singular set lies in a line segment with an endpoint inside the domain must be smooth everywhere, provided the equation has classical density and a Jacobi inequality.

desk verdict A genuinely new proof mechanism and a new single-side theorem, but the Savin stability step is asserted rather than verified; worth refereeing after the gaps are addressed. read the letter →

arxiv 2411.17133 v2 pith:EJSFCPI3 submitted 2024-11-26 math.AP

Ravi Shankar This is my paper · ORCID
classification math.AP MSC 35J6035B6535D4035J96
keywords fullynonlinearellipticequationsviscositysolutionssingularsetremovabilityJacobiinequalityclassicaldensitydoublingMonge-AmpèreequationspecialLagrangian
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

This paper establishes a removability result for singularities of viscosity solutions to a broad class of fully nonlinear elliptic equations. It proves that if the set where a solution fails to be twice differentiable is contained in a line segment with one endpoint inside the domain, then that set is empty: the solution is smooth everywhere. The equations covered are those with two structural properties, classical density and a Jacobi inequality, and they include the Monge-Ampère, minimal surface, and special Lagrangian equations. The proof is short and unifies earlier results: a doubling inequality based on a one-dimensional Green's function forces a Jacobi quantity to be controlled by its boundary values, and a small perturbation stability theorem then upgrades uniform convergence to smooth convergence away from the singularity. The same mechanism removes singular sets satisfying a geometric single-side condition, such as a singularity touching the boundary at one point.

What carries the argument

The key mechanism is a doubling inequality for the Jacobi quantity $b_k = b(D^2 u_k, Du_k)$ on a box $B' = \{a_i \le x_i \le b_i\}$. With $\eta(x_1) = 1/(b_1-x_1) - 1/(b_1-a_1)$, the inequality says $\sup_{B'} \eta(x_1) b_k(x) \le r^{-1} \sup_R b_k(x)$, where $R$ is the boundary of $B'$ excluding the 'bad' side. It is proved by a maximum-principle contradiction: at an interior maximum, ellipticity and the Jacobi inequality force $0 > -\eta''/\eta^2 \ge 0$. The single-side condition makes the exceptional side the only side that can touch the singular set, so $R$ lies in the smooth region; a small perturbation stability theorem then provides uniform bounds on $R$, and the higher bounds from the Jacobi inequality propagate smoothness inward.

What would settle it

Exhibit a viscosity solution of a fully nonlinear elliptic equation with classical density and a Jacobi inequality whose singular set is exactly a half-line segment with one endpoint inside the domain, for example $\{x_1 \ge 0, x_2=x_3=0\}$ in dimension three; Theorem 1.2 predicts no such solution exists.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: if $u \in C(B_1(0))$ is a viscosity solution of $F(D^2u,Du)=0$ with classical density and a Jacobi inequality, and if $\mathrm{sing}(u)$ is contained in a line segment with an endpoint in $B_1(0)$, then $\mathrm{sing}(u)=\emptyset$. Classical density means every viscosity solution is a uniform limit of smooth solutions; the Jacobi inequality means a positive function $b(D^2u,Du)$ satisfies $F^{ij}b_{ij} \ge 2 F^{ij}b_i b_j/b$ along smooth solutions, and local bounds on $b$ imply all higher derivative bounds. Theorem 1.4 generalizes the conclusion: if $\mathrm{sing}(u)$ intersects only one side of a closed box, then $\mathrm{sing}(u)$ is empty inside that box. The Monge-Ampère theorem (Theorem 1.1) is a corollary, and the minimal surface and special Lagrangian equations are shown to satisfy the hypotheses.

Load-bearing premise

The proof rests on the stability claim that near every point where the limit solution is smooth, the approximating smooth solutions have uniformly controlled derivatives, so estimates on the approximations transfer to the limit.

Editorial extensions

If this is right

  • Monge-Ampère viscosity solutions cannot have a singular set contained in a line segment with an endpoint inside the domain; in particular half-line singularities are removable.
  • Minimal surface and special Lagrangian equations admit a unified proof of half-line singularity removal, based only on the Jacobi inequality and classical density rather than on equation-specific Hessian or gradient estimates.
  • For any equation satisfying the two structural hypotheses, any singularity that intersects only one side of a box is removable inside that box; iterating removes 'keyhole' singularities whose closure meets the boundary only once.
  • The proof yields interior smoothness of the limit solution from uniform bounds on the approximations, so the regularity conclusion is $C^{2,\alpha}$ (and higher) at points previously allowed to be singular.

Reading between the lines

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

  • The same doubling framework might extend to equations that only have an almost-Jacobi inequality or lack classical density, if the perturbation stability step can be replaced by a weaker compactness argument; this would widen the class beyond Monge-Ampère, minimal surface, and special Lagrangian.
  • The proof's reliance on a one-dimensional Green's function suggests a route to quantitative versions: the doubling inequality bounds the Jacobi quantity by boundary data, so one could hope for explicit modulus of continuity or derivative bounds near a removable segment.
  • A natural test case is whether half-plane singularities are removable in higher dimensions; the paper notes this is unclear, and the doubling method as written does not seem to handle it, which would be worth probing.
  • The single-side condition is essentially a transversality condition at the boundary; comparing with known full-line singular solutions suggests that some geometric restriction of this kind is necessary, and the method may help identify the optimal condition.
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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

3 major / 6 minor

Summary. The paper gives a short proof of removability of singularities for viscosity solutions of fully nonlinear elliptic PDEs that satisfy two structural hypotheses: classical density (existence of uniformly convergent smooth approximating solutions) and a Jacobi inequality with the associated higher-order estimates. The main theorem, Theorem 1.4, states that if sing(u) is confined to one side of a box, then u is smooth in the interior of the box; Theorems 1.1–1.3 derive the corresponding segment and half-line cases for the Monge-Ampère equation and for the general equations, with the minimal surface and special Lagrangian equations as examples. The proof has four steps: (1) a geometric reduction of the singular set to a single side of a box; (2) a doubling inequality for b(D^2u,Du) obtained by comparing with a one-dimensional Green's function and using the Jacobi inequality; (3) an appeal to Savin's small perturbation theorem to pass from uniform convergence of smooth approximants to local C^{2,α} convergence on the regular set; and (4) higher-order estimates from the uniform bound on b. The paper is explicit that the statements for Monge-Ampère, minimal surface, and special Lagrangian equations are known results, and that its contribution is a new, unified proof.

Significance. If completed, the paper's method is of genuine interest: the doubling inequality in Step 2 is elegant and the contradiction at (2.3) is valid; the single-side formulation in Theorem 1.4 and the keyhole iteration idea in Example 1.1 are appealing; and the paper is candid that the target statements are known for its main examples, citing the classical sources (Caffarelli; Bombieri–De Giorgi–Miranda; Warren–Yuan). The general conditional theorem is a clean framework, and Remark 1.6 lists several recent equations with Jacobi-type inequalities to which the method might plausibly extend. The proof is short and readable. However, the load-bearing stability step (Step 3 and the Appendix) is not completed as written: the passage from uniform convergence to C^{2,α}_loc convergence on the regular set requires regularity of the limit solution u that is exactly what the theorem aims to establish. Because this step is essential for the uniform constant A in (2.5), the central proof is currently incomplete; the paper's value rests on closing this gap, either by a full stability argument or by a strengthened hypothesis.

major comments (3)
  1. [Step 3 and §3 (Appendix)] The transition from uniform convergence u_k → u to local C^{2,α} convergence on the regular set is not justified. The stability proposition in the Appendix is stated under the hypothesis that u is smooth on the subdomain Ω, but Step 3 applies it with Ω a neighborhood of R inside sing(u)^c, where u is only known to be twice differentiable. The linearized operator F(M,p,x) = G(D^2u(x)+M, Du(x)+p) − G(D^2u(x),Du(x)) is not known to satisfy Savin's hypotheses — C^2 in (M,p) uniformly in x, uniform strict ellipticity near (M,p)=0, and F(0,0,x)=0 — on that set, because D^2u and Du are not known to be bounded or continuous there. Consequently the claim 'u_k → u in C^{2,α}_loc(sing(u)^c)' and the uniform bound sup_R b_k ≤ A in (2.5) are not established; the argument is circular in that it assumes the smoothness of u near R that Step 4 is supposed to produce. Even granting smoothness of u on Ω, the one-paragraph proof of Proposition 3.1 sketches rather than verifies the uniform-in-k rescaling, the flatness threshold of Savin's theorem, and the covering argument. This is the load-bearing step of the proof and needs either a full stability argument under the weaker hypothesis or an explicit strengthening of the hypotheses.
  2. [Step 1 (proof of Theorem 1.4)] Step 1 uses 'Since sing(u) is closed' to conclude that the shrunken box B′ has R = ∂B′ \ L0 compactly contained in sing(u)^c. For the generality claimed in Theorems 1.2 and 1.4 — any equation with classical density and a Jacobi inequality — closedness of the complement of the twice-differentiability set is not a consequence of Definition 1.1 and is not proved in the paper; the remark in §1.4 about Savin's theorem and Alexandrov-type results is an assertion rather than an argument. The property is known for the three model equations (Pogorelov for Monge-Ampère, and full interior regularity for minimal surface and special Lagrangian), but the conditional theorems as stated need closedness either proved from the hypotheses or added explicitly.
  3. [§2 (opening) and §1.3 (Theorems 1.1–1.2)] The reduction 'We prove Theorem 1.4, since the others follow from this one' is not demonstrated for Theorems 1.1 and 1.2. When sing(u) is contained in a line segment with an endpoint in B1(0), the segment generically meets the boundary of any box B ⊂ B1 in two points on opposite sides, so the single-side hypothesis of Theorem 1.4 is not automatically satisfied; the half-line case advertised in the abstract requires a further argument (such as a last-singular-point or iteration argument along the segment, along the lines of Example 1.1), and none is supplied. The claimed reductions should be written out.
minor comments (6)
  1. [§2, Step 2 (Eqs. (2.2)–(2.3))] The Jacobi-inequality computation in (2.2) is written for (1/η) after cancelling the positive factor W from the test function φ = W/η, but the cancellation is not displayed; writing the computation for φ explicitly would make the contradiction in (2.3) easier to verify.
  2. [§2, Step 4] The final sentence 'Varying x0, this shows sing(u) ∩ (B′)^0 = ∅, hence sing(u) ∩ B0 = ∅' is not justified, because B′ is a strictly smaller box than B (its right side is at b1 − r); the proof should run the argument for a family of boxes (for instance with r → 0) or otherwise cover B^0, since the conclusion for B^0 does not follow from the conclusion for a single (B′)^0.
  3. [Definition 1.1] The clause 'or ∂F(M,p)/∂M > 0 at all points (M,p) = (D^2u(x), Du(x))' restates the preceding strict ellipticity condition, and the 'or' is confusing; in addition, the inequality ∂F/∂M > 0 should be stated as positive definiteness of the matrix (F^ij).
  4. [Step 1] The phrase 'the smooth, open set sing(u)^c' calls the set of twice-differentiability points the smooth set before smoothness is proved; using a neutral term such as 'regular set' (or proving the smoothness first) would avoid prejudging the conclusion of Step 4.
  5. [Abstract and Remark 1.4] There are typos, including 'viscos ity' in the Abstract and 'follow follow' in Remark 1.4; also, the reference [K87] appears in the bibliography but is not cited in the text.
  6. [Theorem 1.4 / §1.3] The proof would benefit from a precise definition of 'intersects only one side of B' as 'sing(u) ∩ ∂B is contained in the relative interior of a single side', since Step 1 uses exactly this stronger form.

Circularity Check

1 steps flagged · score 6.0 of 10

Step 3's Savin convergence assumes the local smoothness it is meant to prove.

  1. self definitional [Section 2, Steps 1 and 3; Section 3, Appendix]
    "Since sing(u) is closed, we can find a slightly smaller box B′ satisfying the same hypotheses... Thus, R is compactly contained in the smooth, open set sing(u)^c = B1(0) \ sing(u). ... By uniform convergence uk → u in B1(0) and Savin [S07, Theorem 1.3], it follows that uk → u locally smoothly in the smooth set, or in C^{2,α}_{loc}(sing(u)^c)."

    The paper defines sing(u) as the complement of the set where u is merely twice differentiable, not as the complement of the smooth set. Step 1 then relabels sing(u)^c as the “smooth, open set,” and Step 3 uses that label to invoke Savin and conclude uk→u in C^{2,α}_{loc}(sing(u)^c). The Appendix’s linearization F(M,p,x)=G(D²u+M,Du+p)−G(D²u,Du) is C∞ and Savin-eligible only if u is already C² (in fact C∞) in x on Ω; the Appendix assumes exactly “u ... smooth on a subdomain Ω.” That smoothness of u on sing(u)^c is never established—it is a stronger version of the theorem’s conclusion. The uniform bound A in (2.5), on which Step 4’s interior regularity rests, is therefore obtained by assuming local C^{2,α} regularity of u on R rather than deriving it.

full rationale

Most of the paper is not circular: the doubling inequality (2.4) is derived in Step 2 directly from the Jacobi inequality and a one-dimensional Green’s function; the Jacobi inequalities for the concrete equations are cited from external sources (GT77, WY14, Y23); and the self-citations (Sh24, ShY23, SY24) only contextualize the doubling method, with the needed inequality re-derived in the text. No fitted parameters or empirical predictions are involved. The circularity is concentrated in Step 3. The paper defines sing(u) as the complement of twice-differentiability, then labels sing(u)^c the “smooth, open set” and uses that label to apply Savin’s small-perturbation theorem, concluding uk→u in C^{2,α}_{loc}(sing(u)^c). The Appendix makes the hidden premise explicit: the linearized operator F(M,p,x)=G(D²u+M,Du+p)−G(D²u,Du) is C∞ and Savin-eligible only if u itself is C∞ (or at least C²) in x on Ω, and the Appendix assumes “u ... smooth on a subdomain Ω.” That premise is never verified for Ω=sing(u)^c; it is a stronger version of the regularity the theorem is proving. The uniform bound A in (2.5) — and hence Step 4’s interior regularity — therefore rests on assuming the local C^{2,α} regularity of u on R. Step 1’s “Since sing(u) is closed” is the same borrowed partial-regularity conclusion for general equations. This is a genuine circular dependency in the written proof, though not a collapse of the underlying theorem: the gap is repairable by a two-step Savin/localization argument around a twice-differentiability jet of u, and the main results are known from other work. Score 6 reflects partial circularity in the derivation chain, not a verdict on the truth of the theorems.

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

The proof's core is self-contained, but it rests on four external pillars: Savin's theorem, Jacobi inequalities, higher Schauder-type estimates, and classical density. No free parameters are fitted and no new entities are introduced; the Jacobi function b is a known tool in the cited literature.

assumptions (4)
  • standard math Savin's small perturbation theorem [S07, Theorem 1.3]: small viscosity solutions of uniformly elliptic fully nonlinear equations are C^{2,α} with universal estimates.
    Invoked in Step 3 and the Appendix to pass from uniform convergence to smooth convergence of the approximations on the regular set; statement specialized in Proposition 3.1 but not proved.
  • domain assumption Jacobi inequalities for the three model equations: Monge-Ampère b = det(I+D^2u)^{1/(2n)} [Y23]; minimal surface b = sqrt(1+|Du|^2) [GT77]; special Lagrangian b = (1 + lambda_max(D^2u)^2)^{epsilon(n)/2} [WY14].
    The doubling inequality (2.4) is derived from (1.5); these cited inequalities are input assumptions, not proved here.
  • domain assumption Higher-order estimates (1.6): a bound on b(D^2u, Du) implies interior bounds on all derivatives, via Evans-Krylov or De Giorgi-Nash as cited in Remarks 1.3-1.5.
    Step 4 uses (1.6) to convert the doubling bound into smoothness; it is a defining property in Definition 1.1 and is verified externally for the examples.
  • domain assumption Classical density: every viscosity solution is a uniform limit of smooth solutions, obtained by smooth approximation of boundary data and the comparison principle, citing [CNS84], [K84], [GT77], [CNS85], [Y04], [B20].
    Step 2 introduces the approximating sequence u_k; this property is one of the two hypotheses of Theorem 1.2.

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Pith. "Pith review of Removing singularities for fully nonlinear PDEs." pith.science (2026). https://pith.science/paper/EJSFCPI3

@misc{pith2026241117133,
  author       = {Pith},
  title        = {Pith review of: Removing singularities for fully nonlinear PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EJSFCPI3}},
  note         = {Machine review of arXiv:2411.17133}
}
read the original abstract

We show removability of half-line singularities for viscosity solutions of fully nonlinear elliptic PDEs which have classical density and a Jacobi inequality. An example of such a PDE is the Monge-Amp\`ere equation, and the original proof follows from Caffarelli 1990. Other examples are the minimal surface and special Lagrangian equations. The present paper's quick doubling proof combines Savin's small perturbation theorem with the Jacobi inequality. The method more generally removes singularities satisfying the single side condition.

Discussion (0). Continue with ORCID to comment.

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