{"id":"e97275e7-b9e7-48a1-b414-45163c496cd7","arxiv_id":"2508.12595","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Fractional p-Laplace equations on Reifenberg flat domains are shown to satisfy fine boundary regularity for solutions and gradients, claimed new even for the linear fractional Laplacian.","lead":"This paper proves boundary regularity estimates for solutions to fractional p-Laplace equations on rough domains that are flat only in the Reifenberg sense, going beyond Lipschitz boundaries. It extends results previously known for smoother domains or local equations to nonlocal, nonlinear equations, and the authors state the results are new even in the linear fractional case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified from abstract; correctness cannot be assessed without full text.","rationale":"The reader correctly identified the Reifenberg flatness threshold as the load-bearing premise and set UNVERDICTED because the full text was unavailable. My stress-test cannot find a concrete internal inconsistency or a specific hidden assumption from the abstract alone; any attempt would be speculation. The honest assessment is that no significant objection can be raised without the proof. The reader's weakest-assumption analysis is partially aligned: the flatness condition is indeed the key geometric hypothesis, but the reader frames it as a potential failure mode, whereas I see only an unverifiable premise. Therefore I recommend no change to the verdict, and I propose a concrete verification step to be performed once the full text is obtained.","tokens_in":615,"tokens_out":2045,"duration_ms":28533,"concrete_test":"Obtain the full text and inspect the statement of the main boundary regularity theorem (likely Theorem 1.1 or 1.2). Verify that the flatness parameter δ appears as a hypothesis with a threshold that is universal with respect to the solution, depending only on n, s, p, and appropriate data norms, and that the proof does not require choosing δ after the solution is known. If the threshold is data-independent and the estimates are quantitative, the central claim is internally sound on this dimension; if the threshold depends on the solution itself, the theorem's applicability collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract-only version provides no theorem statements, proof outline, or estimate constants. The central claim—fine boundary regularity for nonhomogeneous fractional p-Laplace equations on Reifenberg flat domains, new even in the linear case—cannot be checked for hidden assumptions or mathematical errors. The most load-bearing premise is the Reifenberg flatness hypothesis: if the main theorem requires a flatness parameter δ below a threshold that depends on the solution or the nonhomogeneous term in an unquantified way, the result may become vacuous. However, we have no evidence of such a defect. This is an information limitation, not an identified flaw. In good faith, we cannot substantiate a concrete technical objection from the abstract alone.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript (arXiv:2508.12595) studies nonhomogeneous fractional p-Laplace equations on bounded domains that are Reifenberg flat, a class strictly more general than Lipschitz domains. The abstract claims several fine boundary regularity results for solutions and their gradients under a sufficient flatness assumption, and states that each result is new even in the linear case. No theorem statements, proof outlines, or estimates are included in the available text; the paper is presented only as an abstract.","tokens_in":742,"tokens_out":1212,"duration_ms":16445,"significance":"If the claims are correct, the paper would be a significant contribution to the boundary regularity theory of nonlocal nonlinear equations, extending results beyond Lipschitz boundaries to Reifenberg flat domains. The statement that the results are new even for the linear fractional Laplacian would make the contribution broadly relevant. However, because only the abstract is available, there is no way to verify the core theorems, the precise assumptions, or the proofs. The significance assessment is conditional on the full manuscript substantiating the claims.","major_comments":[{"comment":"The manuscript contains only an abstract. No theorem statements, proof outline, or estimates are provided, so the central claim of boundary regularity cannot be verified. In particular, the 'sufficient flatness assumption' is not quantified. If the required Reifenberg flatness parameter depends on unquantified data such as the solution or the nonhomogeneous term, the result could become narrow or vacuous. This is a load-bearing point that needs precise statement in a full version.","section":"Abstract"},{"comment":"The novelty claim ('new even in the linear case') is unsupported by any references or a literature comparison in the available text. While this is not a mathematical error, it is essential for assessing the contribution. A full manuscript should state the precise relation to prior work on fractional Laplacians in Reifenberg flat and Lipschitz domains.","section":"Abstract"}],"minor_comments":[{"comment":"The abstract uses 'sufficient flatness assumption' without stating the threshold or its dependence on the equation parameters. Adding a one-sentence clarification (e.g., δ < δ0(n, p, ...)) would help readers even at the abstract level.","section":"General"}],"recommendation":"uncertain","confidential_remarks":"The submission as provided to the referee is abstract-only. This is an information limitation rather than an identified mathematical flaw. I cannot recommend acceptance or rejection without the full text. The editor may wish to obtain the complete manuscript before assigning a final decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is an abstract-only submission, so I can't vouch for the math. What I can say is that the claim is clear and, if true, a real step: boundary regularity for fractional p-Laplace equations on Reifenberg flat domains, with the authors asserting the linear case is genuinely new. That is a checkable, falsifiable claim and it aligns with the program that has been running for local p-Laplace systems over the last two decades. The abstract doesn't overreach: it says 'under a sufficient flatness assumption,' which is the standard caveat for Reifenberg-type results. The novelty claim is explicit even in the linear case, which is a good sign because it sets a high bar the authors are willing to be held to. No fitted parameters, no invented constants, no self-referential definitions are visible from the abstract. The soft spot is not in what the paper says but in what the review can see. There are no theorem statements, no proofs, no constants. The load-bearing premise is the flatness threshold: whether the parameter must be smaller than some function of p, n, and the right-hand side, and whether that threshold is quantified. The abstract says 'sufficient,' which is honest but leaves the main theorem's content unverifiable. The citation pattern is also uncheckable from here; the claim that the linear case is new depends on a literature search only a referee can do. I don't see a concrete technical objection to raise. This is a routine information limitation, not a defect. If the full paper is as clean as the abstract, it deserves publication in a good regularity journal. If the flatness threshold is pathologically small, it might end up near-vacuous – but I'd need to see the paper to know. That's the one thing I'd ask a referee to check carefully. Who is this for? Specialists in nonlocal PDE regularity, anyone working on fractional p-Laplace or on Reifenberg domain techniques. It's not a broad-audience paper. Recommendation: yes, send this to peer review. It's a substantive claim with a specific novelty assertion, and the only way to resolve whether it holds is to have a qualified referee read the proofs. I'd advise the editor to ask the referee to verify the linear-case novelty and to check that the flatness threshold does not depend on the solution in a vacuous way.","headline":"Plausible and important claim, but abstract-only means the math is unverified; worth a referee's time.","tokens_in":1208,"tokens_out":1940,"would_cite":false,"duration_ms":22657,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fine boundary regularity holds for fractional p-Laplace equations on domains rougher than Lipschitz.","keywords":["fractional p-Laplace","nonlocal equations","Reifenberg flat domains","boundary regularity","gradient estimates","nonsmooth domains","fractional Laplacian"],"falsifier":"Take a bounded Reifenberg flat domain whose flatness parameter satisfies the theorem's assumption but whose boundary is self-similar, solve the nonhomogeneous fractional $p$-Laplace equation with a smooth right-hand side, and test whether the weak solution extends continuously to a boundary point; a discontinuous trace at any such point would refute the claim.","tokens_in":516,"feed_emoji":"📏","tokens_out":6473,"duration_ms":76592,"temperature":0.7,"pith_summary":"This paper takes up nonhomogeneous fractional $p$-Laplace equations on bounded domains whose boundaries are uniformly well approximated by hyperplanes at every scale—the Reifenberg flat condition—and proves that solutions are continuous up to the boundary, with fine quantitative estimates on their gradients. Such domains can be far rougher than Lipschitz, so the result stretches boundary regularity for nonlocal nonlinear equations into a genuinely nonsmooth class. The authors assert that each boundary regularity estimate is new even when the equation is linear, i.e., for the fractional Laplacian.","feed_headline":"Rough flat boundaries keep nonlocal solutions regular","feed_subtitle":"Fractional p-Laplace equations gain boundary continuity on non-Lipschitz Reifenberg flat domains.","key_machinery":"Reifenberg flatness of the domain, a scale-invariant condition that the boundary is within a small Hausdorff distance of a hyperplane at every point and scale. This condition lets the proof flatten the boundary locally and transfer interior energy estimates for the fractional $p$-Laplace operator up to the boundary. The operator itself is the nonlocal, nonlinear singular integral whose regularity theory is being extended.","core_discovery":"On a bounded domain whose boundary is Reifenberg flat—uniformly well approximated by hyperplanes at every scale—every weak solution of a nonhomogeneous fractional $p$-Laplace equation is continuous up to the boundary, and its gradient satisfies quantitative estimates there, provided the flatness parameter is small enough relative to the data. The geometry is not required to be Lipschitz; the boundary may be genuinely nonsmooth. The authors state that these boundary regularity results are new even when the equation reduces to the linear fractional Laplacian.","pith_inferences":["The same mechanism likely extends to other translation-invariant nonlocal operators, since only scale-invariant boundary geometry is used.","For the linear case ($p=2$), the boundary regularity has probabilistic meaning: solutions of stable-process Dirichlet problems inherit continuity at the boundary on Reifenberg flat domains.","The flatness condition may be near-optimal: if the threshold is dropped, boundaries with oscillating shapes likely destroy boundary regularity, so the result might identify a quantitative phase transition."],"forward_implications":["Boundary continuity and gradient bounds now hold on Reifenberg flat domains, which include boundaries with fractal or cusp-like roughness that are not Lipschitz.","The linear fractional Laplacian gains the same boundary regularity on such domains, a previously open case.","The quantitative nature of the estimates makes them usable in compactness and existence arguments for nonlinear nonlocal problems on rough domains.","The flatness threshold serves as an explicit smallness condition that can be checked for concrete domains."],"supporting_citations":[],"fun_headline_variants":["Fractional p-Laplace gains boundary control on Reifenberg flat domains","Reifenberg flatness tames nonlocal equations at the boundary","Non-Lipschitz boundaries admit continuity for fractional p-Laplace","New boundary estimates for fractional p-Laplace on Reifenberg flat sets","Boundary regularity for fractional p-Laplace even on rough flat domains"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The boundary must be uniformly approximable by hyperplanes within a flatness parameter small enough to meet a threshold that depends on the equation's data; if the boundary's flatness is only coarse, the boundary regularity conclusion may fail.","fun_headline_variants_meta":{"raw":{"variants":["Fractional p-Laplace gains boundary control on Reifenberg flat domains","Reifenberg flatness tames nonlocal equations at the boundary","Non-Lipschitz boundaries admit continuity for fractional p-Laplace","New boundary estimates for fractional p-Laplace on Reifenberg flat sets","Boundary regularity for fractional p-Laplace even on rough flat domains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000621,"raw_usage":{"total_tokens":2615,"prompt_tokens":541,"completion_tokens":2074,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":285,"completion_tokens_details":{"reasoning_tokens":1981}},"tokens_in":285,"tokens_out":2074,"duration_ms":14588,"temperature":1.0,"reasoning_tokens":1981,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:24:20.092841+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a bounded Reifenberg flat domain whose flatness parameter satisfies the theorem's assumption but whose boundary is self-similar, solve the nonhomogeneous fractional $p$-Laplace equation with a smooth right-hand side, and test whether the weak solution extends continuously to a boundary point; a discontinuous trace at any such point would refute the claim.","supporting_citations":[],"review_version":1}