Global fractional Sobolev regularity for fully nonlinear elliptic equations

Claudemir Alcantara; Makson Santos

REVIEW 5 major objections 5 minor 17 references

Global fractional Sobolev regularity for fully nonlinear elliptic equations

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Pith's one-line read This paper proves that bounded viscosity solutions of uniformly elliptic fully nonlinear equations belong to W^{γ,p}(Ω) up to the boundary for every γ<1+ε, with quantitative estimates, assuming only uniform ellipticity and a small…

desk verdict The global boundary regularity result is attractive, but the proof has a central gap in Lemma 3.7 that breaks the boundary decay chain, so it needs referee work before it can be trusted. read the letter →

arxiv 2411.15311 v1 pith:OZOFOUEY submitted 2024-11-22 math.AP

Claudemir Alcantara , Makson Santos This is my paper · ORCID

classification math.AP MSC 35B6535J6035J1535J25

keywords fractionalSobolevregularityfullynonlinearellipticequationsviscositysolutionsC^{1α}-conesLaplacianboundaryW^{γp}estimatesuniformellipticity

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 establishes global fractional Sobolev regularity for viscosity solutions of fully nonlinear elliptic equations of the form F($D^{2}$u,Du,u,x)=f, under only uniform ellipticity and a small oscillation condition on F in the x-variable. The main results show that solutions belong to $W^{{γ,p}}$ up to the boundary for every γ<1+ε, with ε∈(0,α0), where α0 is the interior $C^{{1,α0}}$ Hölder exponent of F-harmonic functions. The proof works by showing that a solution extended by zero near a flat boundary is a weak solution of a fractional Laplacian equation of order σ/2 with σ<1+ε, so that the fractional regularity follows from known theory for the fractional Laplacian. This matters because it yields differentiability of order strictly greater than one at the boundary without convexity or concavity assumptions on the operator, a regime where classical $C^{{2,α}}$ or $W^{{2,p}}$ boundary estimates are not available.

What carries the argument

The proof is carried by $C^{{1,α}}$-cones, functions of the form ψ(x)=ℓ(x)±(M/2)|x−x0|^{1+α} with ℓ affine. For each scale, the sets G_M(u,Ω) of points that can be touched from above and below by such cones, and their complements A_M, are studied through a Calderón–Zygmund decomposition; the aperture function θ(x)=inf{M: x∈G_M} measures how much cone opening is needed at x. The main mechanism is the estimate $θ^{{1+α}}$∈L^p, obtained from the decay of |A_{M^k}| with k, which follows from the approximation lemma (Proposition 3.1). Once θ is integrable, the second-difference quotient $Δ_h^{{1+α}}$ũ is bounded by θ, and the singular integral I_{σ/2} representing (−Δ)^{σ/2} is controlled in L^p, producing the fractional Laplacian equation.

What would settle it

If a uniformly elliptic operator F satisfying A1–A4 admits a bounded viscosity solution on the half-ball with zero boundary data that fails to lie in $W^{{1+δ,p}}$ for some δ>0, or if the constant-coefficient limit equation in Proposition 3.1 has a solution that is not $C^{{1,α0}}$ up to the flat boundary, the boundary regularity claim collapses.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that viscosity solutions to uniformly elliptic fully nonlinear equations inherit fractional-diffusion structure up to the boundary: if u solves F($D^{2}$u,x)=f in the upper half-ball with zero Dirichlet data, then the zero-extension ũ is a weak solution of (−Δ)^{σ/2}ũ=g for some g∈L^p and every σ<1+ε. This yields u∈$W^{{γ,p}}$(B^+_{1/2}) for all γ<σ, with the estimate ‖u‖_{$W^{{γ,p}}$} ≤ C(‖u‖_{L^∞}+‖f‖_{L^p}). The boundary statement is then transported to general $C^{{1,1}}$ domains and non-zero $W^{{2,p}}$ Dirichlet data by flattening the boundary and subtracting the boundary values, giving the global estimate of Theorem 2.2. The regularity is quantitative and holds in the same range p>d−ε0 that is known for interior $W^{{1+ε,p}}$ estimates.

Load-bearing premise

The approximation step in Proposition 3.1 assumes that the limit of solutions with vanishing right-hand sides is $C^{{1,α0}}$ on the closed half-ball, which requires boundary $C^{{1,α0}}$ regularity for constant-coefficient fully nonlinear equations with zero data on the flat boundary.

Editorial extensions

If this is right

Viscosity solutions of uniformly elliptic fully nonlinear equations possess fractional differentiability of order greater than one up to the boundary, without convexity or concavity assumptions.
The fractional Sobolev norm is controlled by the L^∞ norm of the solution, the L^p norm of the source, and the W^{2,p} norm of the boundary data, with a constant depending only on universal parameters.
The result extends to gradient-dependent operators satisfying the structural condition A5, in the range p>d−ε0.
Since W^{1+ε,p} embeds into C^{1,α} for suitable p, the order 1+ε is essentially optimal, consistent with known counterexamples to C^{1,β} regularity.
The boundary regularity is achieved by reducing the original problem to a fractional Laplacian equation, so known fractional Calderón–Zygmund theory applies directly.

Reading between the lines

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

A natural test of the method is whether the exponent ε0 in Theorem 2.2 can be improved to the Escauriaza exponent for the boundary problem, since the paper uses the interior value; failure for p between d−ε0 and the boundary-optimal range would indicate the boundary argument is not sharp.
The same C^{1,α}-cone setup could be adapted to Neumann or oblique boundary conditions, where the flat-boundary fractional Laplacian structure would take a different form because the extension beyond the boundary is not zero.
The paper leaves implicit that the approximation lemma's boundary C^{1,α0} assumption might be supplied by existing boundary regularity theory under additional structure; if that structure is necessary, the result would hold only for a subclass of uniformly elliptic operators.

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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

Dear colleague,

You asked about arXiv:2411.15311. Here's my read.

The paper aims to prove global W^{γ,p} regularity with γ>1 up to the boundary for uniformly elliptic fully nonlinear equations without convexity, including gradient dependence and variable coefficients. That would be a genuine extension of the interior result in [11] and would improve on the boundary results in [17] and [13]. The theorem statements are new and worth taking seriously.

What is good: the cone method from [11] is adapted to the boundary in a natural way, and the authors correctly identify that the main work is to get a distributional estimate for the aperture function θ near the boundary. The writing is clear, the assumptions are standard (A1-A5), and the interior appendix extends [11] to gradient-dependent operators in a plausible way. If the proof were solid, this would be a nice paper.

Where it falls down: the boundary decay chain rests on Lemma 3.7, and I don't think the proof works as written. The step "w = (v-h)/2 is a viscosity solution of G" is not justified. The approximating h from Proposition 3.1 is only C^{1,α0}_{loc}, not C^2, and it solves a constant-coefficient limit equation F∞(D^2h)=0, not the original operator \tilde F(D^2h,x)=0. Without a C^2 solution of the same operator, the viscosity solution argument cannot be made. This is not a minor technicality: it is the mechanism that transfers the decay estimate for A_t(w) to the original u, and without it the proof of θ∈L^p near the boundary collapses. The stress-test note on this point is accurate.

There are smaller issues too. The singular integral in Theorem 2.1's proof has the wrong middle term (-2u(y) instead of -2u(x0)); that looks like a typo, but as written the L^p estimate by θ doesn't follow. The range of σ in the statement is loose but not contradictory. The approximation in Proposition 4.1 and the appendix are sketched with references to [17] and [12], and the appendix adds assumptions A6-A7 that are not in the main theorems; the authors should clarify whether the main result depends on them.

If I were refereeing, I would ask the authors to fix Lemma 3.7 or supply a different argument for the boundary decay. The result is plausible and likely fixable, but the current manuscript doesn't carry the proof.

Who should look at it: researchers in fully nonlinear elliptic regularity, especially those working with the C^{1,α}-cone method. It deserves a serious referee, not a desk reject, because the claimed theorem is important and the flaws are specific and potentially repairable.

I wouldn't cite it in my own work until the proof is repaired.

Best, [You]

Referee Report

5 major / 5 minor

Summary. The paper claims global fractional Sobolev regularity up to the boundary for viscosity solutions of uniformly elliptic fully nonlinear equations with measurable ingredients and possibly gradient-dependent structure. The strategy is to prove a boundary version of the interior result of Pimentel–Santos–Teixeira [11]: the solution truncated to the half-ball is shown to be a weak solution of a fractional Laplacian equation of order σ/2, from which W^{γ,p} boundary regularity follows for γ<1+ε, with ε∈(0,α0). Theorem 2.1 states the flat-boundary, zero-Dirichlet case for F(D^2u,x)=f; Corollary 2.1 and Theorem 2.2 extend this to non-zero boundary data on C^{1,1} domains and to operators depending on Du and u, assuming smallness of the x-oscillation of the operator and an L^p source with p>d−ε0. The proof uses C^{1,α}-cones, a Calderón–Zygmund decomposition at the boundary, an approximation lemma, and a reduction to a fractional Laplacian via a singular-integral bound.

Significance. If the main results were fully established, they would constitute a notable extension of fractional Sobolev regularity to the boundary for fully nonlinear elliptic equations without convexity assumptions, complementing the interior results of [11] and improving on the W^{1,p} and W^{2,p} boundary estimates of Winter [17] under very mild structural hypotheses. The paper has a clear and attractive strategy: it reduces the boundary problem to a fractional-Laplacian representation and then uses known regularity for the fractional Laplacian. The paper also carefully states the C^{1,α}-cone machinery and the Calderón–Zygmund decomposition at the boundary. However, several load-bearing steps in the written proof are incomplete or unjustified, most notably the boundary approximation lemma (Proposition 3.1), the viscosity-solution reduction in Lemma 3.7, and the explicit omission of the mollification argument in Proposition 4.1. These gaps currently prevent the main claims from being considered proven.

major comments (5)

[Section 3, Proposition 3.1] The proof of the approximation lemma is not established up to the boundary. The contradiction argument produces a limit u∞ that solves the constant-coefficient equation F∞(D^2u∞)=0 in the open half-ball B^+_{12√d}, and the paper then asserts u∞∈C^{1,α0}(B^+_{12√d}) merely from the fact that F∞ is uniformly elliptic. No boundary C^{1,α0} estimate for the flat-boundary problem is proved or cited, and the zero trace u=0 on {x_d=0} may be lost under locally uniform convergence in the open half-ball. This matters because Lemma 3.7 later uses h from Proposition 3.1 to control the sets A_N(h,B^+_{12√d})∩((Q^{d-1}_1×(0,1))+x0), which requires regularity and trace information up to the flat boundary.
[Section 3, Lemma 3.7] The definition w:=(v−h)/2 and the assertion that w is a viscosity solution of G(M,x)=1/2 \tilde F(2M+D^2h,x)=\tilde f are not justified. The function h obtained from Proposition 3.1 is only C^{1,α0}_{loc} in the open half-ball and solves the limit equation F∞(D^2h)=0, not the equation \tilde F(D^2h)=0 for the same operator for which v is a solution. The viscosity test-function argument requires h to be an admissible perturbation, e.g., h∈C^2 or a classical solution of the same operator; neither condition is supplied. Since Lemma 3.7 feeds into Lemma 3.8 and Proposition 3.2, the boundary decay mechanism for the cone sets is not proven as written.
[Theorem 2.1 and its proof] There is a contradiction between the stated range of σ and the proof. The theorem states that σ∈(0,1+ε), while the proof works with '1<σ<1+α' and Corollary 2.1 concludes u∈W^{γ,p} for γ<σ. If σ can be smaller than 1, the claimed conclusion that solutions have differentiability of order strictly greater than one does not follow. The intended statement is presumably σ∈(1,1+ε), but as written the theorem does not imply the advertised regularity.
[Proof of Theorem 2.1, singular integral] The singular integral is defined with the wrong middle term: I_{σ/2}(v)(x0)=∫[v(x0+y)+v(x0−y)−2v(y)]/|y|^{d+σ}dy, and the same erroneous expression −2\tilde u(y) is used in the estimate that follows. The cone-touch bound gives control of |u(x0+y)+u(x0−y)−2u(x0)|, not of the printed numerator. As written, the L^p bound on the fractional Laplacian of \tilde u is not justified. Even if this is a typo, it appears in the central step and must be corrected and re-verified.
[Section 4, Proposition 4.1] The passage from the gradient-dependent operator F(D^2u,Du,u,x) to the x-dependent operator \tilde F(D^2u,x)=F(D^2u,0,0,x) relies on an approximation argument that the paper explicitly omits ('Although we omit the detailed argument here'). This step is load-bearing for Theorem 2.2: one must show that the mollified operators satisfy Assumption A4 with uniform constants, that the corresponding solutions u_j exist and solve the approximated equations, and that u_j→u weakly in W^{γ,p}; none of this is demonstrated. Without these details, the reduction to Theorem 2.1 is incomplete.

minor comments (5)

[Definition 2.7 and Assumption A4] The notation '≤β_0^d' in A4 is confusing: β_0 is a small constant and d is the dimension; it is likely intended to be a power of the constant, but the dimension in the exponent is not explained.
[Lemma 3.6, proof, second case] In the covering argument, the term |A(u,Ω)∩Q_ε(x_i)| is missing the subscript t; it should read |A_t(u,Ω)∩Q_ε(x_i)|.
[Proof of Theorem 2.2] The symbol φ is overloaded: it denotes the boundary data in the theorem statement but also a function φ∈W^{2,p}(B^+_1) in the flattening argument. The definition of \tilde F also mixes v, φ and ϑ, making the pulled-back equation difficult to follow.
[Remark 2.2 and Appendix, Proposition 5.1] Remark 2.2 refers to Proposition 5.1 as establishing that F-harmonic functions belong to W^{2,p}(B^+_1), but Proposition 5.1 only proves density of W^{2,p} solutions; the actual W^{2,p} estimate is Theorem 5.1, which requires additional assumptions A6 and A7 that are not present in the main theorems. This discrepancy should be clarified.
[Various places] There are numerous typographical issues, including missing spaces and missing superscripts (for example 'u ∈C(B6r√d)' in Lemma 3.1, and 'B1/2' in several places). These do not affect the mathematics but should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the fractional-Laplacian representation is derived from cone-touching and Calderón–Zygmund estimates, not from a fitted or self-referential input.

full rationale

The derivation chain is self-contained in the relevant sense. The aperture function θ is defined through C^{1,α}-cone touchings (Definitions 2.2–2.5), and the proof of Proposition 3.2 obtains θ ∈ L^p by Calderón–Zygmund decomposition and maximal-function estimates (Lemmas 3.5, 3.6, 3.8). Theorem 2.1 then bounds the fractional Laplacian integral I_{σ/2}(ẽ u) pointwise by θ plus an L^∞ term, using the second difference of the touching cone; this is a genuine implication rather than a renaming of the conclusion. No parameter is fitted to data, and no target regularity is assumed in the hypotheses. The paper relies substantially on [11] (Pimentel–Santos–Teixeira), a published external result whose assumptions do not include the present boundary theorem; because one of the present authors is a coauthor of [11], this is a self-citation, but it is not circular: the new content is the boundary estimate, the flat-boundary fractional-Laplacian representation, and the extension to gradient-dependent operators and C^{1,1} domains. The most serious issue is a proof gap, not a circularity: Proposition 3.1 supplies only an interior C^{1,α_0} limit h, while Lemma 3.7 asserts that w=(v-h)/2 is a viscosity solution of an equation containing D^2h; this requires boundary regularity or a different argument and is a correctness concern, not an equivalence of inputs and outputs. The central claim does not reduce to its assumptions by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper's added value is the boundary adaptation of the C^{1,α}-cone method; it does not introduce new physical entities. The load-bearing inputs are standard elliptic regularity theorems plus two extra appendix assumptions A6/A7 that are not in the main hypothesis list.

free parameters (3)

β_0 (A4 smallness) = unspecified; chosen sufficiently small
Assumption A4 requires an average oscillation of F below β0^d; every approximation and density argument uses this smallness, but no quantitative bound is given.
Escauriaza exponent ε_0 = unspecified small positive exponent from [5]
Theorem 2.2 and Remark 2.1 push p to d−ε0 using [5]/[17]; ε0 is universal but never computed.
Proof constants M, t, σ, ε0 in Lemmas 3.3-3.8 and Prop 3.2 = chosen sufficiently large or small; existence only
The Calderón-Zygmund and iteration arguments require choosing M>1, t>1, 0<σ<1, and ε0 with ε0 M^p=1/2; the theorem only needs their existence.

assumptions (5)

standard math Krylov-Safonov/Trudinger: there exists α0∈(0,1) such that solutions of F(D^2u)=0 are C^{1,α0} locally
Used throughout; cited to [8] and [16]; α0 remains unknown.
domain assumption Boundary C^{1,α0} regularity for constant-coefficient F∞(D^2u)=0 in half-ball with zero flat-boundary data
Used in Prop 3.1 to choose h=u∞; not explicitly cited or proved in the paper.
ad hoc to paper A6/A7: the recession operator F* has C^{1,1} estimates and an oscillation property
Introduced in the appendix for Prop 5.1 and Theorem 5.1; not part of A1-A5, and not verified for the operators in the main theorems.
standard math W^{1,p} boundary estimates of Winter [17] and pointwise a.e. differentiability of Caffarelli et al. [3]
Used in Prop 4.1 to construct ilde f∈L^p and to pass from the a.e. equation to the viscosity equation.
standard math C^{1,α}-cone estimates and [11, Lemma 7] from the interior paper
The boundary proof imports the interior decay and touching estimates from [11] (co-authored by Santos); treated as an external theorem.

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Cite this review

Pith. "Pith review of Global fractional Sobolev regularity for fully nonlinear elliptic equations." pith.science (2026). https://pith.science/paper/OZOFOUEY

@misc{pith2026241115311,
  author       = {Pith},
  title        = {Pith review of: Global fractional Sobolev regularity for fully nonlinear elliptic equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZOFOUEY}},
  note         = {Machine review of arXiv:2411.15311}
}

abstract

We investigate fractional regularity estimates up to the boundary for solutions to fully nonlinear elliptic equations with measurable ingredients. Specifically, under the assumption of uniform ellipticity of the operator, we demonstrate that viscosity solutions to a second-order operator satisfy a fractional Laplacian equation. This result implies that the solutions are globally of class $W^{\gamma, p}$, for $\gamma \in (1,2)$, with appropriate estimates. Consequently, these solutions exhibit differentiability of order strictly greater than one, without requiring any additional assumptions regarding the operator, such as convexity or concavity.

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Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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Edgard Pimentel, Makson Santos, and Eduardo Teixeira, Fractional sobolev regularity for fully non- linear elliptic equations , Communications in Partial Diﬀerential Equations 47 (2022), no. 8, 1539– 1558. 21

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Edgard Pimentel and Eduardo Teixeira, Sharp Hessian integrability estimates for nonlinear ellip tic equations: an asymptotic approach , J. Math. Pures Appl. (9) 106 (2016), no. 4, 744–767. MR 3539473

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [11]

8, 1539– 1558

Edgard Pimentel, Makson Santos, and Eduardo Teixeira, Fractional sobolev regularity for fully non- linear elliptic equations , Communications in Partial Diﬀerential Equations 47 (2022), no. 8, 1539– 1558. 21

work page 2022

[2] [17]

Niki Winter, W 2,p and W 1,p -estimates at the boundary for solutions of fully nonlinear , uniformly elliptic equations , Z. Anal. Anwend. 28 (2009), no. 2, 129–164. MR 2486925 Claudemir Alcantara Department of Mathematics Pontiﬁcal Catholic University of Rio de Janeiro (PUC-Rio), 451-900, Rio de Janeiro, Brazil alcantara@mat.puc-rio.br Makson S. Santos D...

work page 2009

[3] [13]

Partial Diﬀerential Equations 39 (2014), no

Luis Silvestre and Boyan Sirakov, Boundary regularity for viscosity solutions of fully nonli near elliptic equations, Comm. Partial Diﬀerential Equations 39 (2014), no. 9, 1694–1717. MR 3246039

work page 2014

[4] [12]

Edgard Pimentel and Eduardo Teixeira, Sharp Hessian integrability estimates for nonlinear ellip tic equations: an asymptotic approach , J. Math. Pures Appl. (9) 106 (2016), no. 4, 744–767. MR 3539473

work page 2016

[5] [1]

Boumediene Abdellaoui, Antonio Fern´ andez, Leonori Tommaso, and Younes Abdelbadie, Global fractional calder´ on-zygmund type regularity, arXiv:2107.06535v3

arXiv

[6] [2]

Luis Caﬀarelli, Interior a priori estimates for solutions of fully nonlinea r equations , Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611

work page 1989

[7] [3]

4, 365–398

Luis Caﬀarelli, Michael Crandall, Maciej Kocan, and Andrzej ´Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Communications on Pure and Applied Mathematics 49 (1996), no. 4, 365–398

work page 1996

[8] [4]

Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369

work page 2012