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$C^{1,\alpha}$ regularity of the solution for the obstacle problem for the linearized Monge-Amp\`ere operator

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Pith's one-line read This paper shows that strong solutions of the obstacle problem for the linearized Monge-Ampère operator are locally $C^{1,\gamma}$ for every $\gamma\in(0,1)$, and that an $L^n$-viscosity solution exists and is unique.

desk verdict New and honest but the central iteration in Lemma 4.2 has a scale-gap that prevents the proof of C^{1,γ} regularity from starting. read the letter →

arxiv 2505.24410 v2 pith:CDE4HXIC submitted 2025-05-30 math.AP

Meng Ji This is my paper
classification math.AP MSC 35J7035R3535D4035J96
keywords obstacleproblemlinearizedMonge-AmpèreoperatorC^{1\gamma}regularityviscositysolutionfreeboundarysectionsdegenerateellipticequationsW^{2n}strong
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 studies the obstacle problem where a function must stay above a given obstacle while solving the linearized Monge-Ampère equation away from contact, with zero boundary values. It proves existence and uniqueness of an $L^n$-viscosity solution, and then proves that if the solution is a strong solution in $W^{2,n}_{\mathrm{loc}}(\Omega)$, its gradient is locally Hölder continuous with every exponent $\gamma<1$, including across the free boundary. This matters because the linearized Monge-Ampère operator can be degenerate, and degeneracy usually threatens regularity; the paper shows that under a strong-solution hypothesis the degeneracy does not prevent gradients from being almost Lipschitz.

What carries the argument

The argument is carried by the sections of $w$, defined as $S_{h,w}(x_0)=\{x\in\Omega: w(x)\le w(x_0)+\nabla w(x_0)(x-x_0)+h\}$, which play the role that balls play for uniformly elliptic operators. After an affine normalization, these sections are comparable to a fixed ball, and the two Gutierrez–Nguyen lemmas quantify how well a normalized small section is approximated by $B_{\sqrt{2}}$. Iterating this normalization produces a sequence of positive-definite matrices $A_k$ with controlled size, leading to the estimate that the gradient grows at most like $r^\alpha$ away from the free boundary. The Harnack inequality for the linearized Monge-Ampère operator and the interior $W^{2,p}$ estimate then convert the growth control into $C^{1,\gamma}$ regularity.

What would settle it

Find a function $w$ satisfying (1.3) and a strong solution $u\in W^{2,n}_{\mathrm{loc}}(\Omega)\cap C(\bar\Omega)$ of the obstacle problem whose gradient fails to be Hölder continuous for some exponent $\gamma<1$ near a free boundary point; this would directly contradict Theorem 1.2. A more local test is to verify numerically or analytically whether the normalized section iteration of Lemma 2.8 preserves the claimed $B_{\sqrt{2}}$ comparability for all iterates, since the proof depends on that recursion.

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Extended reading notes

Core claim

Theorem 1.2 is the central claim: assume the obstacle and domain satisfy the standing conditions (1.3), and let $u\in W^{2,n}_{\mathrm{loc}}(\Omega)\cap C(\bar\Omega)$ be a solution of the obstacle problem. Then $u\in C^{1,\gamma}_{\mathrm{loc}}(\Omega)$ for any $\gamma\in(0,1)$. The proof works by fixing a free boundary point, normalizing the Monge-Ampère section of $w$ at that point, and iterating a geometric rescaling so that the normalized section is always comparable to a fixed ball; this forces the gradient of the normalized solution to be continuous at the free boundary and to grow at most like a power of the distance. Theorem 1.1 supplies uniqueness and existence of an $L^n$-viscosity solution via Perron's method, but the regularity theorem is conditional on that solution being a strong solution in $W^{2,n}$.

Load-bearing premise

The paper assumes, rather than proves, that the solution whose existence it establishes is strong enough to have second derivatives in $L^n$; if that regularity fails, Theorem 1.2 does not apply to the constructed solution.

Editorial extensions

If this is right

  • Any strong $W^{2,n}$ solution has locally Hölder continuous gradient with every exponent $\gamma<1$, including at points where the solution touches the obstacle.
  • The $L^n$-viscosity solution from Theorem 1.1 is unique, so the regularity question reduces to whether that solution is strong in $W^{2,n}$.
  • The free boundary does not create a gradient singularity: near every free boundary point the normalized gradient is continuous and grows at most like a power of the distance.
  • Under the stated assumptions one cannot expect $C^{1,1}$ regularity, since $w$ itself is only known to be in $W^{2,p}$ for $p<\infty$; stronger ellipticity assumptions on $f$ would be needed.

Reading between the lines

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

  • If a future argument establishes $W^{2,n}$ regularity for the $L^n$-viscosity solution built by Perron's method, Theorem 1.2 immediately upgrades that solution to $C^{1,\gamma}$; the current paper does not close that gap.
  • The section-normalization iteration is a general template that may transfer to other degenerate elliptic obstacle problems whose coefficients have comparable section geometry, giving an 'any exponent below 1' conclusion under a strong $W^{2,n}$ hypothesis.
  • The proof leaves open a quantified version: the Hölder exponent $\gamma$ is shown to be arbitrary in $(0,1)$ but the constants are not tracked in terms of $\lambda,\Lambda,n$, so a testable extension is to determine whether the modulus of continuity of $f$ controls the size of the $C^{1,\gamma}$ norm uniformly.
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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

4 major / 5 minor

Summary. The paper studies the obstacle problem for the linearized Monge-Ampère operator L_w u = tr(W D^2 u) ≤ 0, with W the cofactor matrix of D^2 w and w satisfying λ ≤ det D^2 w ≤ Λ. Theorem 1.1 states existence and uniqueness of an L^n-viscosity solution in C(Ωbar) by Perron's method and a comparison principle. Theorem 1.2 claims that any strong solution u ∈ W^{2,n}_{loc}(Ω) ∩ C(Ωbar) is locally C^{1,γ} for every γ ∈ (0,1). The proof strategy is to normalize a section S_h,w(x0) at a free boundary point, iterate two geometric lemmas of Gutiérrez-Nguyen to control the growth of sections, and then combine a boundary-growth estimate with interior regularity in the non-contact set. The paper also contains a smoothing argument (Theorem 3.3) that is intended to justify harmonic replacement in sections, and a remark explaining why C^{1,1} is not expected under the given assumptions.

Significance. If the main regularity theorem were valid, it would be a notable step for obstacle problems for degenerate elliptic operators, connecting section geometry for Monge-Ampère equations to free-boundary regularity. The paper is honest about the conditional nature of Theorem 1.2, and the proof relies on established tools (Caffarelli-Gutiérrez Harnack inequality, Gutiérrez-Nguyen W^{2,p} estimates and section lemmas) rather than on fitted parameters or self-citation. However, the proof of Theorem 1.2 contains a load-bearing gap concerning the quantitative modulus of continuity of f, and the smoothing argument in Theorem 3.3 does not establish its stated boundary conclusion. Because these gaps affect the central claim, the manuscript cannot be accepted in its present form.

major comments (4)
  1. [Proof of Theorem 1.2 (paragraph after Eq. (4.16)) and Eq. (4.7)] The final proof uses the same h0 as the height of the original section S_{h0}(x0) and as the iteration parameter μ in Lemmas 2.7 and 2.8. Lemma 2.7 requires ε ≤ τ0 μ^2 and Lemma 4.2 imposes Eq. (4.7), √ε ≤ C θ h0 log(1/h0). In the normalized section, ε is the oscillation of f(T^{-1}y)/f(x0) over S_{h0}(x0). For merely continuous f this oscillation need not be O(h0^2): by Theorem 2.4 the section has diameter at most C h0^σ, so for example f(x)=1+|x−x0|^{1/2} gives oscillation of order h0^{σ/2}, which is much larger than h0^2 for small h0. Thus the inequalities required to start the iteration cannot be ensured by choosing h0 small. Lemma 4.2's separate choice of a very small section height h does not repair the final theorem, because the normalization bounds (4.17) and the final C^{1,γ} constants depend on the section height; replacing h0 by a much smaller h makes |A| ∼ h^{-1} and the constants blow up.
  2. [Theorems 1.1 and 1.2] Theorem 1.1 constructs only an L^n-viscosity solution u ∈ C(Ωbar), while Theorem 1.2 assumes that u ∈ W^{2,n}_{loc}(Ω) ∩ C(Ωbar). No argument is given that the Perron solution satisfies the W^{2,n} hypothesis, nor is any other existence result for strong solutions provided. Therefore the regularity theorem does not apply to the solution whose existence is established. If the authors intend Theorem 1.2 as a purely conditional regularity statement, this should be stated with much more prominence; if they intend it as the main result of the paper, a bridge from Theorem 1.1 to the strong-solution hypothesis is missing.
  3. [Theorem 3.3] The proof does not establish the boundary condition V = v on ∂B1. The approximating problems are solved in D with boundary data only on ∂D; the limit V in C^{1,α}(B1bar) inherits its values on ∂B1 from the interior solution values V_k, not from the prescribed data v|_{\partial B1}. Consequently the conclusion (3.1) does not follow from the argument. In addition, the assertion that the W^{2,s} bound from Lemma 2.6 is independent of k requires a common modulus of continuity for the approximating determinants f_k; uniform convergence f_k → f alone does not supply such a modulus uniformly in k. This matters because Lemma 4.1 invokes Theorem 3.3 for the harmonic replacement in sections.
  4. [Lemma 4.2, Eq. (4.13)] The first supremum in Eq. (4.13) is over a section S_{h0^{k-1},φ*}(y0), but the subsequent application of Lemma 4.1 and the annulus estimate (4.11) concern sections S_{h0^{k-1},w*}(y0). These are different objects, and the displayed estimate is not connected to the quantity being bounded. Also, the sentence 'δk is a decreasing sequence' is not correct: for small h0, δ2 ≈ C^2√ε/√h0 is larger than δ1 ≈ C√ε/h0. The later closed-form sum for δk does not rely on monotonicity, but the presentation is misleading.
minor comments (5)
  1. [Proof of Theorem 2.4] The line '2^{-(i+1)}t0 ≤ h ≤∈ 2^{-i}t0' contains a typo; it should be 'h ∈ [2^{-(i+1)}t0, 2^{-i}t0]'.
  2. [Theorem 3.3] The word 'Sovolev' should be 'Sobolev' in the sentence 'by Sovolev embedding'.
  3. [Lemma 4.2, Eq. (4.13)] The section S_{h0^{k-1},φ*}(y0) appears to be a typo for S_{h0^{k-1},w*}(y0); please correct and check the supremum carefully.
  4. [Lemma 3.2] The proof of the comparison principle assumes that the infimum of φ = v - u is attained. For L^p-viscosity solutions that are merely continuous this needs justification, or the proof should invoke a standard inf/sup-convolution argument.
  5. [Theorem 1.2, after Eq. (4.17)] The notation must distinguish the height h of the section used for normalization from the iteration parameter h0. In the current text, 'choose h0 small enough' is used for both, which is exactly what makes the modulus-of-continuity issue in the first major comment unavoidable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main regularity theorem is conditional on a strong-solution hypothesis and depends on external regularity tools, not on self-citation or fitted inputs.

full rationale

The paper's central claim, Theorem 1.2, is conditional: it states that if u is a strong solution in W^{2,n}_{loc}(Omega) cap C(overline{Omega}), then u is C^{1,gamma}_{loc} for any gamma in (0,1). This is not a prediction derived from fitted parameters or from the existence result of Theorem 1.1; it is an implication under an explicit, unproved regularity hypothesis. The proof uses external results: Caffarelli-Gutierrez Harnack inequality, Gutierrez-Nguyen interior estimates, and Gutierrez-Nguyen section lemmas (Lemmas 2.7 and 2.8). These are independent literature results, not author self-citations, and their assumptions do not already contain the target conclusion. The smallness condition on epsilon in Lemma 4.2 is imposed by the hypotheses of the external Lemma 2.7, and the exponents alpha and gamma are determined by the constants in the proof, not by fitting to the desired conclusion. The paper does contain a genuine gap: Theorem 1.1 produces an L^n-viscosity solution, while Theorem 1.2 requires W^{2,n}_{loc} regularity, and no bridge is established; additionally, the reviewer's concern that a merely continuous f may not satisfy the quantitative smallness needed to start the iteration is a substantive correctness risk. However, a missing hypothesis or an invalid step in a proof is not circularity unless the conclusion is equivalent to the input by construction. No such equivalence is present here. The only self-referential element is the acknowledgment of the author's supervisor, which is not load-bearing. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof imports standard Monge-Ampere section geometry (John's lemma, section inclusions), Caffarelli-Gutierrez Harnack, Gutierrez-Nguyen W^{2,p} and section-iteration lemmas, plus the unproved W^{2,n}_{loc} strong-solution hypothesis for the main theorem. No new physical entities or fitted constants are introduced.

assumptions (6)
  • standard math John's lemma: any convex body contains an ellipsoid and is contained in its n-dilation
    Used in Lemma 2.1 to normalize sections and the domain, enabling the iteration estimates.
  • standard math Caffarelli strict convexity: if det D2w is bounded between positive constants and w=0 on the boundary, then w is strictly convex
    Invoked after (1.3) to guarantee sections S_{h,w}(x0) have nonempty interior.
  • standard math Harnack inequality for L_w (Theorem 2.5)
    External result from Caffarelli-Gutierrez used in Lemma 4.1 to control the oscillation of harmonic replacements.
  • standard math Interior W^{2,p} estimate for L_w (Lemma 2.6)
    External result from Gutierrez-Nguyen used in Theorem 3.3 and Lemma 4.4 to get Holder continuity of gradients in the non-contact region.
  • standard math Section approximation lemmas of Gutierrez-Nguyen (Lemmas 2.7 and 2.8)
    These geometric lemmas drive the inductive control of normalized sections in Lemma 4.2; their hypotheses require det D2w* close to 1 and normalized convex domains.
  • domain assumption u in W^{2,n}_{loc}(Omega) intersect C(Omega bar) is a strong solution of the obstacle problem
    This is the premise of Theorem 1.2. The paper proves only an L^n-viscosity solution exists; the strong regularity is assumed rather than derived.

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Pith. "Pith review of $C^{1,\alpha}$ regularity of the solution for the obstacle problem for the linearized Monge-Amp\`ere operator." pith.science (2026). https://pith.science/paper/CDE4HXIC

@misc{pith2026250524410,
  author       = {Pith},
  title        = {Pith review of: $C^1,\alpha$ regularity of the solution for the obstacle problem for the linearized Monge-Amp\`ere operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDE4HXIC}},
  note         = {Machine review of arXiv:2505.24410}
}
abstract

In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Amp\`ere operator: \begin{align*} \begin{cases} &u\geq\varphi \text{\quad in } \Omega &L_{ w}u=\tr( W D^{2}u)\leq 0 \text{\quad in } \Omega &L_{ w}u= 0 \text{\quad in } \{u>\varphi\} &u=0 \text{\quad on } \partial\Omega, \end{cases} \end{align*} where $ W=(\det D^{2} w) D^{2} w^{-1}$ is the matrix of cofactor of $D^{2} w$, $w$ satisfies $\lambda \leq \det D^{2} w \leq \Lambda$ and $ w=0$ on $\partial \Omega$, $\varphi$ is the obstacle with at least $C^{2}(\bar{\Omega})$ smoothness, $\Omega$ is an open bounded convex domain. We show the existence and uniqueness of a viscosity solution by using Perron's method and the comparison principle. Our primary result is to prove that the solution exhibits local $C^{1,\gamma}$ regularity for any $\gamma \in (0,1)$, provided that it is a strong solution in $W^{2,n}_{\text{loc}}(\Omega)$.

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