Boundary H\"older regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

Adriano Prade

REVIEW 3 major objections 3 minor 1 cited by

Boundary H\"older regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

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Pith's one-line read This paper proves that weak solutions of the fractional Laplacian with zero exterior condition are Hölder continuous up to the boundary on sufficiently flat Reifenberg flat domains, for every fractional exponent s and every Hölder…

desk verdict A well-written extension of the Reifenberg boundary Hölder program that fails on a load-bearing barrier estimate: the assertion v0 ≥ d^s contradicts v0 ≤ 1. read the letter →

arxiv 2501.14639 v1 pith:CPIUOSL3 submitted 2025-01-24 math.AP

Adriano Prade This is my paper · ORCID

classification math.AP MSC 35R1135B6535D30

keywords fractionalLaplacianboundaryHölderregularityReifenbergflatdomainsnonlocalABPmaximumprincipleRieszpotentialsweaksolutionsbarrierfunctionsintegro-differentialequations

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 proves that weak solutions of the fractional Laplacian $(-\Delta)^s$ with zero exterior condition are Hölder continuous up to the boundary on Reifenberg flat domains, provided the flatness parameter is small enough. For every $s\in(0,1)$ and every Hölder exponent $\alpha\in(0,s)$, the solution satisfies $|u(x)|\le C\,d_\Omega(x)^\alpha$ and finally $u\in C^\alpha(\overline{\Omega})$ with the norm controlled by $\|f\|_{L^\infty(\Omega)}$. The result extends to all fractional powers a boundary regularity statement that was previously known for smooth, $C^1$, and Lipschitz domains, and in the nonlocal setting only for $s=1/2$. The proof is an iterative rescaling argument that uses a barrier built from Riesz potentials and a nonlocal version of the ABP maximum principle.

What carries the argument

The machinery is the pair consisting of a barrier function $v_0$ and the nonlocal ABP maximum principle. The barrier is $v_0=1-I_{2s}(\tilde B_1)^{-1}v_{\tilde B_1}$, where $v_{\tilde B_1}$ is the potential of the equilibrium measure of a small ball $\tilde B_1$; it solves $(-\Delta)^s v_0=0$ outside $\tilde B_1$, vanishes on $\tilde B_1$, is $C^s$ with $v_0\le C_H\eta^s$ near $\tilde B_1$, and is claimed to grow like $d(\cdot,\tilde B_1)^s$ far away. Rescaled copies $v_k$ of this barrier are superimposed on $u$, and the nonlocal ABP estimate $\sup (v_k-u)^- \le C_{\mathrm{ABP}}\operatorname{diam}(\Omega\cap B_{\lambda^k})\|f\|_{L^\infty}$ — a maximum principle bounding the negative part of a supersolution in terms of the domain diameter and the $L^\infty$ norm of the right-hand side — converts the equation into a pointwise bound at each inductive step. The choice $m(s-\alpha)>s$ determines the flatness condition $\eta=\lambda^{(s+m\alpha)/s}$ that makes the induction close.

What would settle it

Compute $v_0$ for a concrete case such as $n=2$, $s=1/2$ and evaluate $v_0(Re_1)$ for $R>4$: the claimed inequality $v_0(x)\ge 4^{-s}|x|^s$ fails because $v_0\le 1$ while $|x|^s>1$, which would invalidate the exterior comparison on which the inductive step rests.

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Extended reading notes

Core claim

The central claim is Theorem 2: for any $\alpha\in(0,s)$, once the flatness parameter $\eta$ is below a threshold $\eta_0(n,s,\alpha)$, the weak solution of $(-\Delta)^s u=f$ in $\Omega$ with $u=0$ on $\mathbb{R}^n\setminus\Omega$ belongs to $C^\alpha(\overline{\Omega})$ and satisfies $\|u\|_{C^\alpha(\overline{\Omega})}\le C(n,s,\alpha)\|f\|_{L^\infty(\Omega)}$. The proof first establishes the boundary growth estimate $|u(x)|\le C M d_\Omega(x)^\alpha$ by an induction over the nested balls $B_{\lambda^k}$: at each step the rescaled barrier $v_k$ is compared with $u$ using the nonlocal ABP principle, giving the control on the next shell $B_{\lambda^{k+m}}$; once the growth estimate is available, interior regularity upgrades it to full Hölder continuity. The constant $M$ is $\|u\|_{L^\infty} + C_{\mathrm{ABP}}\|f\|_{L^\infty}$.

Load-bearing premise

The proof assumes the barrier $v_0$ grows at least like the $s$-th power of the distance to a small ball, in particular $v_0(x)\ge 4^{-s}|x|^s$ for $|x|>1$; if that growth fails, the exterior comparison $v_k-u\ge0$ in the induction step collapses and the argument does not close.

Editorial extensions

If this is right

Every weak solution with bounded right-hand side is Hölder continuous at every boundary point, with any exponent $\alpha<s$, on domains that are sufficiently flat in the Reifenberg sense.
The estimate $|u(x)|\le C\|f\|_{L^\infty} d_\Omega(x)^\alpha$ holds, so solutions vanish at the boundary at least as fast as the $\alpha$-th power of the distance.
The result closes the gap between local elliptic equations and the fractional Laplacian on Reifenberg flat domains: the previously known nonlocal case $s=1/2$ now extends to all $0<s<1$.
The proof yields a quantitative flatness threshold $\eta_0(n,s,\alpha)$; any domain flatter than this threshold satisfies the same boundary regularity estimate with constants independent of the geometry.

Reading between the lines

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

The same barrier-plus-ABP strategy should transfer to integro-differential operators whose kernels are comparable to $|y|^{-n-2s}$; the proof only uses the equation's scaling, the comparison principle, and the ABP-type estimate, so the announced generalization to more general nonlocal operators is a natural next step.
Tracking constants in the choice $\eta=\lambda^{(s+m\alpha)/s}$ suggests that the admissible flatness $\eta_0$ shrinks as $\alpha$ approaches $s$, matching the intuition that finer boundary regularity demands flatter domains.
The boundary growth proved here is $|u(x)|\le C d_\Omega(x)^\alpha$ for every $\alpha<s$; it is natural to expect the optimal exponent is $s$ itself, as in the half-space boundary behaviour, and that a sharper barrier or a boundary Harnack argument might reach it.

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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

The paper has a real idea but a proof gap that looks fatal as written. The author extends Goldman–Novaga–Ruffini's s=1/2 potential-function estimate to all s in (0,1) and to distributional solutions with continuous right-hand side, adapting Lian–Zhang's iterative ABP argument to the nonlocal setting. The writing is clear, the strategy is sensible, and the acknowledgment of the literature is honest. For a proceedings contribution, the exposition is above average.

The problem is in the barrier construction. Immediately before (7) the paper claims v0(x) ≥ d(x,B̃1)^s, citing the maximum principle, and uses this to derive the growth bound v0(x) ≥ 4^{-s}|x|^s for |x|>1. But earlier it correctly states v0∈[0,1]. Since v0 is the normalized capacitary potential 1 − I_{2s}(B̃1)^{-1} v_{B̃1}(x), it vanishes on B̃1, is increasing, and tends to 1 at infinity; it never exceeds 1. The function d(x,B̃1)^s is unbounded, so the inequality is false for large |x|. This is not a minor typo: the induction uses (7) to obtain vk(x) ≥ CM λ^{(k−1)α} λ^{(j−k)s} for x ∈ Ω\B_{λ^k}, then compares it with u(x) ≤ CM λ^{(j−1)α}. In the critical annulus j=k, the claim requires vk to be bounded below by a positive constant independent of distance to B_{λ^k}, but the true v0 is only O(d^s) and vanishes at B̃_{λ^k}. Thus the exterior comparison vk−u ≥ 0 in (12) is not established, and the nonlocal ABP maximum principle cannot be applied. The central theorem is unsupported.

The result may well be true, and the approach may be repairable with a different barrier or a sharper local estimate, but the current proof does not deliver it. I would not accept the paper as is. It deserves a serious referee only because the underlying question is meaningful and the method is worth checking; the referee should focus on the barrier estimate. My recommendation: major revision or reject, with the burden on the author to fix the barrier growth or explain a local substitute.

Referee Report

3 major / 3 minor

Summary. The paper studies (-Delta)^s u = f in a bounded connected (eta,r0)-Reifenberg flat domain with u = 0 outside Omega. It claims Theorem 2: for every alpha in (0,s), if eta <= eta0(n,s,alpha), then the weak solution is C^alpha up to the boundary with ||u||_{C^alpha(Omega)} <= C(n,s,alpha) ||f||_infty. The proof introduces a normalized equilibrium potential v0 of a small ball, rescales it at scales lambda^k, and uses the Guillen-Schwab nonlocal ABP maximum principle to propagate a boundary Holder bound by induction.

Significance. If the theorem were established, it would be a natural extension of boundary Holder regularity for the fractional Laplacian to Reifenberg flat domains, complementing the C^1, C^{1,gamma} and Lipschitz results. The paper is clearly organized and makes honest use of imported tools, and the induction combined with the nonlocal ABP principle is an interesting strategy. However, the proof hinges on a barrier estimate that is false; as written the central argument collapses. The theorem may still be true, but the present manuscript does not prove it.

major comments (3)

[Barrier construction, Eq. (7)] The estimate v0(x) >= d(x,Btilde1)^s stated immediately before (7) is incompatible with the definition of v0. Since v0(x) = 1 - I_{2s}(Btilde1)^{-1} v_{Btilde1}(x) and v_{Btilde1}(x) = O(|x|^{-(n-2s)}) at infinity, v0 is bounded and tends to 1; the paper itself states v0 in [0,1]. But d(x,Btilde1)^s >= (|x|-1/2)^s, which exceeds 1 for all sufficiently large |x|. The cited maximum principle can at most yield a local boundary estimate of order d^s near Btilde1, not the global growth bound (7).
[Proof of Theorem 2, Eq. (12)] Because (7) is false, the proof that vk - u >= 0 on (Omega cap B_{lambda^k})^c is not established, and the failure is load-bearing: vk <= 4^s C M lambda^{(k-1)alpha} tends to 0 as k -> infinity, while the induction hypothesis permits u to be of order M on the far part of Omega \ B_{lambda^k}. No rescaling of this bounded equilibrium potential can dominate an arbitrary bounded solution on the whole exterior of B_{lambda^k} for large k. Hence the exterior comparison in (12) fails, Guillen and Schwab's ABP maximum principle cannot be applied, and the induction proving (8) collapses.
[End of proof of Theorem 2] The final step claims a global estimate (2) with C = C(n,s,alpha) by covering Omega with finitely many balls of radius 1/2. The proof does not show that the number of such balls is bounded by a function of n, s and alpha alone; in general it depends on the size and geometry of Omega. Unless the domain size is normalized or the statement allows C to depend on Omega, the uniform estimate does not follow from the preceding local argument.

minor comments (3)

[After Eq. (15)] The notation x is used both for a point in B_{3/4} and for the boundary point realizing d_Omega(x); please use different symbols such as y and x0.
[Barrier construction, paragraph before Eq. (6)] The phrase 'radially symmetric and monotone increasing around the point' should be replaced by 'radially symmetric and radially nondecreasing with respect to that point'.
[Throughout] The display '( -Delta)^s' and the word 'H\"older' contain broken diacritics and spacing in the source; please ensure the final PDF renders these correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the proof uses an external ABP maximum principle, a potential-theoretic barrier, and independent solution-equivalence results; no fitted parameter is relabeled as a prediction.

full rationale

The central estimate (8) is obtained by an induction that combines the fractional Laplacian equation with a barrier v0 built from the equilibrium measure of a fixed ball and with the Guillen-Schwab nonlocal ABP maximum principle. All of these ingredients are imported external results: potential theory for Riesz energy, the ABP estimate, and the equivalence between weak/distributional and viscosity solutions. No constant in the proof is fitted to u, to the domain Ω, or to the target Hölder exponent α in a way that would force the conclusion. The constants M and C depend on ‖u‖_∞ and ‖f‖_∞, and the L∞ bound is cited as an independent known lemma. The only self-citation, [8], is an announcement of upcoming work and plays no load-bearing role. The skeptical concern about the growth estimate v0(x) ≥ d(x,B̃1)^s, and its apparent inconsistency with the earlier statement v0 ∈ [0,1], is a possible correctness gap in the barrier construction; it is not a circularity, because that inequality is not an assumption equivalent to the theorem and no quantity is renamed as a prediction. Accordingly, no circular step is identified.

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

The paper introduces no fitted parameters. Its proof imports the Guillen-Schwab ABP theorem, potential theory for equilibrium measures, and distributional/viscosity equivalence from the literature, which is legitimate background. The load-bearing extra assumption is the unsupported and false lower bound for v0. There are no invented physical entities.

assumptions (4)

ad hoc to paper The equilibrium potential v0 satisfies v0(x) >= d(x,Btilde1)^s for all relevant x.
Asserted before Eq. (7); used to derive the growth estimate (7). It conflicts with the stated bound v0 in [0,1] when d(x,Btilde1) > 1, so the axiom is not available.
domain assumption The barrier ball lambda^k Btilde1 is contained in the complement of Omega cap B_{lambda^k}.
Used to conclude (-Delta)^s vk = 0 on Omega cap B_{lambda^k}; the proof states this inclusion without deriving it from Reifenberg flatness.
domain assumption Guillen-Schwab nonlocal ABP maximum principle applies to the pair (vk - u) with the stated hypotheses.
Imported from [4]; the proof verifies formal hypotheses but relies on the exterior comparison that the faulty barrier estimate was supposed to supply.
domain assumption Distributional and viscosity solutions of the linear fractional Laplacian coincide in the needed class.
Quoted from [1, Lemma 3.4.13]; standard and not at issue.

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Cite this review

Pith. "Pith review of Boundary H\"older regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle." pith.science (2026). https://pith.science/paper/CPIUOSL3

@misc{pith2026250114639,
  author       = {Pith},
  title        = {Pith review of: Boundary H\"older regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPIUOSL3}},
  note         = {Machine review of arXiv:2501.14639}
}

abstract

For $0<s<1$, we consider the nonlocal equation $(-\Delta)^s u = f$ over a Reifenberg flat domain $\Omega$ with $f \in C({\overline{\Omega}})$ and null Dirichlet exterior condition. Given $\alpha \in (0,s)$, we prove that weak solutions are $\alpha$-H\"older continuous up to the boundary when the flatness parameter is small enough. The main ingredients of the proof are an iterative argument and a nonlocal version of the ABP maximum principle.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains
math.AP 2025-02 conditional novelty 6.0 of 10

For operators comparable to the fractional Laplacian of order 2s, solutions with zero exterior data on Reifenberg flat domains are C^{s-ε} up to the boundary.

Reference graph

Works this paper leans on

8 extracted references · 5 canonical work pages · cited by 1 Pith paper

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Integro-diﬀerential elliptic equations

Fern´andez-Real, X., and Ros-Oton, X. Integro-diﬀerential elliptic equations. Progress in Mathematics, Birkh¨ auser Cham(2024)

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Existence and stability for a non-local isoperimetric mode l of charged liquid drops

Goldman, M., Novaga, M., and Ruffini, B. Existence and stability for a non-local isoperimetric mode l of charged liquid drops. Archive for Rational Mechanics and Analysis 217 (2015), 1–36

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Rigidity of the ball for an isoperimetric problem with stron g capacitary repulsion

Goldman, M., Novaga, M., and Ruffini, B. Rigidity of the ball for an isoperimetric problem with stron g capacitary repulsion. Journal of the European Mathematical Society (2024)

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Guillen, N., and Schw ab, R. W. Aleksandrov–Bakelman–Pucci type estimates for integro-d iﬀerential equations. Archive for Rational Mechanics and Analysis 206 (2012), 111–157

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Landkof, N. S. Foundations of modern potential theory , vol. 180. Springer, 1972

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Boundary regularity for the Poisson equation in Reifenberg -ﬂat domains

Lemenant, A., and Sire, Y. Boundary regularity for the Poisson equation in Reifenberg -ﬂat domains. In Geo- metric Partial Diﬀerential Equations proceedings (2013), Springer, pp. 189–209

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Boundary H¨ older regularity for elliptic equations on Reif enberg ﬂat domains

Lian, Y., and Zhang, K. Boundary H¨ older regularity for elliptic equations on Reif enberg ﬂat domains. arXiv preprint arXiv:1812.11354 (2018)

arXiv 2018
[8]

Boundary regularity for nonlocal elliptic equations over R eifenberg ﬂat domains

Prade, A. Boundary regularity for nonlocal elliptic equations over R eifenberg ﬂat domains. In preparation. CMAP, ´Ecole Polytechnique, 91120 P alaiseau, France. Email address : adriano.prade@polytechnique.edu

work page

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Integro-diﬀerential elliptic equations

Fern´andez-Real, X., and Ros-Oton, X. Integro-diﬀerential elliptic equations. Progress in Mathematics, Birkh¨ auser Cham(2024)

2024

[2] [2]

Existence and stability for a non-local isoperimetric mode l of charged liquid drops

Goldman, M., Novaga, M., and Ruffini, B. Existence and stability for a non-local isoperimetric mode l of charged liquid drops. Archive for Rational Mechanics and Analysis 217 (2015), 1–36

work page 2015

[3] [3]

Rigidity of the ball for an isoperimetric problem with stron g capacitary repulsion

Goldman, M., Novaga, M., and Ruffini, B. Rigidity of the ball for an isoperimetric problem with stron g capacitary repulsion. Journal of the European Mathematical Society (2024)

2024

[4] [4]

Guillen, N., and Schw ab, R. W. Aleksandrov–Bakelman–Pucci type estimates for integro-d iﬀerential equations. Archive for Rational Mechanics and Analysis 206 (2012), 111–157

work page 2012

[5] [5]

Landkof, N. S. Foundations of modern potential theory , vol. 180. Springer, 1972

work page 1972

[6] [6]

Boundary regularity for the Poisson equation in Reifenberg -ﬂat domains

Lemenant, A., and Sire, Y. Boundary regularity for the Poisson equation in Reifenberg -ﬂat domains. In Geo- metric Partial Diﬀerential Equations proceedings (2013), Springer, pp. 189–209

work page 2013

[7] [7]

Boundary H¨ older regularity for elliptic equations on Reif enberg ﬂat domains

Lian, Y., and Zhang, K. Boundary H¨ older regularity for elliptic equations on Reif enberg ﬂat domains. arXiv preprint arXiv:1812.11354 (2018)

arXiv 2018

[8] [8]

Boundary regularity for nonlocal elliptic equations over R eifenberg ﬂat domains

Prade, A. Boundary regularity for nonlocal elliptic equations over R eifenberg ﬂat domains. In preparation. CMAP, ´Ecole Polytechnique, 91120 P alaiseau, France. Email address : adriano.prade@polytechnique.edu

work page