{"id":"4a63088d-97dd-4dbf-8d03-766b6b64d343","arxiv_id":"2501.14639","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Boundary Hölder regularity for the fractional Laplacian over Reifenberg flat domains is claimed for all 0<s<1, but the proof relies on a barrier estimate that contradicts the boundedness of the barrier.","lead":"This paper claims that weak solutions of the fractional Laplacian, zero outside a Reifenberg flat domain, are Hölder continuous up to the boundary for every fractional order s. The proof's key barrier inequality conflicts with the paper's own boundedness statement, so the central claim is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof relies on the barrier inequality v0(x) ≥ d(x,B̃1)^s, which contradicts the same paper's v0∈[0,1] and is false for large |x|; this inequality is exactly what forces vk−u≥0 outside B_{λ^k} before applying ABP.","rationale":"I read Theorem 2 as the central claim: weak solutions to (−Δ)^s u=f with zero exterior data are C^α up to the boundary on sufficiently flat Reifenberg domains, for every s∈(0,1) and α∈(0,s). The proof is an iterative barrier construction combined with the nonlocal ABP maximum principle. For this strategy to work, the rescaled barrier vk must lie above u on the complement of Ω∩B_{λ^k}; otherwise ABP is not applicable. The manuscript obtains this by deriving vk(x)≥CMλ^{(k−1)α}|x/λ^k|^s on Ω\\B_{λ^k} from inequality (7). The reader correctly identified a contradiction: the barrier is asserted to satisfy both v0∈[0,1] and v0≥d(x,B̃1)^s. I checked this in the actual construction: v_{B̃1} is the potential of the equilibrium measure of total mass 1, and v0=1−I2s^{-1}v_{B̃1} is the standard capacitary potential normalized to vanish on B̃1 and approach 1 at infinity; hence 0≤v0≤1. For large |x|, the lower bound d^s exceeds 1, so (7) is impossible. This is not a matter of conventions or of a missing constant. The problematic inequality is used exactly in the verification of (12), especially in the adjacent dyadic annulus j=k, where it supplies a uniform positive lower bound vk≥CMλ^{(k−1)α} just outside B_{λ^k}. The true behavior of v0 near ∂B̃1 is O(d^s), which tends to 0 near that boundary. Consequently vk−u≥0 on (Ω∩B_{λ^k})^c is unproved, and the ABP application is unsupported. No other part of the proof supplies the missing comparison: the L∞ bound only gives |u|≤M, and the induction hypothesis is precisely what the proof is trying to establish. There is no machine-checked verification and no numerical or independent confirmation offered. The reader's high-confidence REJECT is therefore appropriate, and my stress-test does not change that verdict.","tokens_in":5496,"tokens_out":7700,"duration_ms":69756,"concrete_test":"Independently verify (7) at x=10 e_n: since v0≤1, the left side is at most 1, while d(x,B̃1) ≥ 9.75, so the right side exceeds 1 for every s>0; hence (7) is false as stated. Then check what replaces it in the induction: with the true asymptotic v0(x)=1−I2s|x|^{-(n−2s)}(1+o(1)), the step j=k gives only vk≤4^sCMλ^{(k−1)α}+o(λ^{(k−1)α}) on Ω\\B_{λ^k}, which does not dominate u in Ω∩(B_{λ^{k−1}}\\B_{λ^k}) unless additional boundary decay of u is already known. Thus the exterior comparison (12) fails in general.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is the assertion, immediately before (7), that v0(x) ≥ d(x,B̃1)^s \"by the maximum principle,\" combined with the earlier statement that v0∈[0,1]. Since v0 is the normalized capacitary potential 1 − I2s(B̃1)^{-1}v_{B̃1}(x), it vanishes on B̃1 and tends to 1 at infinity; in particular v0≤1 everywhere. For x=R e_n with R>1, d(x,B̃1)^s ≥ (R−1/4)^s >1 for sufficiently large R, so v0≥d^s is impossible. The intended inequality can at best be a local boundary estimate near B̃1, not a global growth bound. The induction needs exactly the global form: for x∈Ω\\B_{λ^k} with |x|∈[λ^j,λ^{j−1}), the proof uses (7) to obtain vk(x) ≥ CMλ^{(k−1)α+(j−k)s} and then compares it with u(x)≤CMλ^{(j−1)α}. In the adjacent annulus j=k this lower bound is vk≥CMλ^{(k−1)α}, i.e. a uniform positive barrier just outside B_{λ^k}; the true v0 is only O(d^s), which vanishes there. Therefore the exterior comparison vk−u≥0 in (12) is not established, the nonlocal ABP maximum principle cannot be applied, and the induction proving (8) collapses. This is an internal inconsistency in the argument, not a matter of disagreement with current consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5808,"tokens_out":14578,"duration_ms":145696,"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":[{"comment":"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).","section":"Barrier construction, Eq. (7)"},{"comment":"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.","section":"Proof of Theorem 2, Eq. (12)"},{"comment":"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.","section":"End of proof of Theorem 2"}],"minor_comments":[{"comment":"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.","section":"After Eq. (15)"},{"comment":"The phrase 'radially symmetric and monotone increasing around the point' should be replaced by 'radially symmetric and radially nondecreasing with respect to that point'.","section":"Barrier construction, paragraph before Eq. (6)"},{"comment":"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.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The main theorem may be true, but the proof's central barrier estimate is impossible and the proposed rescaling cannot work with a bounded barrier. I recommend rejection. If the author finds a genuinely different barrier or reformulates the exterior comparison, the result could be reconsidered; the covering/uniform-constant issue should also be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.","headline":"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.","tokens_in":6340,"tokens_out":2827,"would_cite":false,"duration_ms":26136,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35B65","35D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["fractional Laplacian","boundary Hölder regularity","Reifenberg flat domains","nonlocal ABP maximum principle","Riesz potentials","weak solutions","barrier functions","integro-differential equations"],"falsifier":"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.","tokens_in":5240,"feed_emoji":"📐","tokens_out":15287,"duration_ms":119137,"temperature":0.7,"pith_summary":"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.","feed_headline":"Flat domains yield Hölder continuity for every fractional s","feed_subtitle":"For every s between 0 and 1, weak solutions become Hölder continuous at the boundary when the domain is flat enough.","key_machinery":"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.","core_discovery":"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 $\\|u\\|_{L^\\infty} + C_{\\mathrm{ABP}}\\|f\\|_{L^\\infty}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the nonlocal toolkit: weak/distributional solution equivalence, the $L^\\infty$ bound for weak solutions, and the interior regularity used to upgrade boundary growth to Hölder continuity.","marker":"[1]"},{"why":"Provides the properties of the barrier (radial symmetry, monotonicity, the system $(-\\Delta)^s v_0=0$ outside the ball, $v_0=0$ on it, and the small $C^s$ constant near the ball).","marker":"[2]"},{"why":"Yields the nonlocal ABP maximum principle that converts the equation for $v_k-u$ into a pointwise bound at each rescaling.","marker":"[4]"},{"why":"Guarantees existence and uniqueness of the equilibrium measure for the Riesz energy of the ball $\\tilde B_1$, from which the barrier is built.","marker":"[5]"},{"why":"Supplies the iterative rescaling strategy with a flat barrier and the ABP principle that this proof adapts to the nonlocal setting.","marker":"[7]"}],"fun_headline_variants":["Fractional Laplacian gains Hölder boundary regularity on flat domains","ABP principle unlocks boundary Hölder bounds for fractional s","Small flatness yields C^alpha up to boundary for all 0<s<1","Nonlocal ABP proves boundary Hölder continuity on Reifenberg flat sets","For every s, flat domains force Hölder solutions at the boundary"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Laplacian gains Hölder boundary regularity on flat domains","ABP principle unlocks boundary Hölder bounds for fractional s","Small flatness yields C^alpha up to boundary for all 0<s<1","Nonlocal ABP proves boundary Hölder continuity on Reifenberg flat sets","For every s, flat domains force Hölder solutions at the boundary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1360,"prompt_tokens":876,"completion_tokens":484,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":389}},"tokens_in":492,"tokens_out":484,"duration_ms":5435,"temperature":1.0,"reasoning_tokens":389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:58:50.581583+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Existence and stability for a non-local isoperimetric mode l of charged liquid drops","cited_arxiv_id":null,"evidence_quote":"Provides the properties of the barrier (radial symmetry, monotonicity, the system $(-\\Delta)^s v_0=0$ outside the ball, $v_0=0$ on it, and the small $C^s$ constant near the ball)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Yields the nonlocal ABP maximum principle that converts the equation for $v_k-u$ into a pointwise bound at each rescaling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Guarantees existence and uniqueness of the equilibrium measure for the Riesz energy of the ball $\\tilde B_1$, from which the barrier is built."}],"review_version":1}