{"id":"a2dd956e-4e8a-4ef5-b9c8-e8f7fc454009","arxiv_id":"2607.28436","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the s-fractional p-Laplacian with f in L^q, solutions are globally α-Hölder up to the boundary with α almost p'(s−N/pq) or s, and u/d_Ω^s is Hölder when q>N/s.","lead":"Solutions of the fractional p-Laplacian Dirichlet problem with L^q forcing stay Hölder continuous up to the boundary, with almost-optimal exponents. The work also gives weighted fine boundary regularity when the forcing is sufficiently integrable, extending the linear theory to the nonlinear nonlocal setting.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly identifies the C^{1,1} requirement as the principal modelling limitation and correctly notes that the theorems as stated remain valid under that hypothesis. The proofs of global Hölder continuity (via iterative Campanato estimates on harmonic extensions, Lemmas 5.1–5.2) and of fine boundary regularity (via approximation from L^\\infty, weighted monotonicity and barriers, Section 4) are detailed and close the estimates needed for the claimed ranges of q and \\alpha. The almost-optimality examples (1.4 and Appendix A) further corroborate that the thresholds are sharp. No load-bearing analytic gap appears; the verdict ACCEPT with high confidence is therefore left unchanged.","tokens_in":37598,"tokens_out":437,"duration_ms":8201,"concrete_test":"Independently re-derive the key variance bound (5.7) in the proof of Lemma 5.2 for the model case p=2 (linear fractional Laplacian) and check that the resulting iteration recovers the known linear exponents of (1.4) from [15,21]; agreement confirms the nonlinear iteration does not introduce a spurious loss.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorems 1.3 and 1.5 are supported by a complete, self-contained argument that stays inside standard nonlocal elliptic technique (Campanato oscillation, (s,p)-harmonic extensions, barriers from Prop. 3.1, fractional Hardy, and the known L^\\infty theory of [17,19]). The C^{1,1} hypothesis flagged by the reader is stated explicitly, used only where the exterior-ball condition and unique nearest-point projection are needed (Lemmas 2.10–2.11, barrier construction, weighted estimates), and is not hidden. The undetermined exponent in the fine-boundary result and the failure to reach the endpoint \\bar \\alpha are openly acknowledged limitations, not gaps that invalidate the stated theorems. No internal inconsistency or missing estimate that would break the claimed range of \\alpha was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the Dirichlet problem for the s-fractional p-Laplacian with homogeneous exterior data and reaction f in L^q(\\Omega), q > N/(ps). Theorem 1.3 asserts global Hölder continuity of the unique weak solution u up to the boundary: u \\in C^\\alpha(\\Omegā) for every \\alpha \\le s with \\alpha < p'(s - N/(pq)) when N/(ps) < q \\le N/s, and for \\alpha = s when q > N/s, with the natural estimate in terms of \\| f\\|_{L^q}^{1/(p-1)}. Theorem 1.5 asserts that if q > N/s then the quotient u/d_\\Omega^s admits a C^\\alpha extension to \\Omegā for some (undetermined) \\alpha \\in (0,s], again with a corresponding estimate. The proofs combine Campanato oscillation estimates, (s,p)-harmonic extensions, barrier constructions, fractional Hardy inequalities, and separate monotonicity formulas in the degenerate and singular regimes, building on known interior regularity and the bounded-reaction boundary theory.","tokens_in":37719,"tokens_out":877,"duration_ms":24898,"significance":"The results give a nearly complete nonlinear analogue of the linear scheme (1.4) for unbounded reactions, filling a clear gap between the interior Hölder theory (Brasco–Lindgren–Schikorra, Garain–Lindgren) and the global/fine-boundary theory available only for L^\\infty data. The almost-optimality examples (1.4 and Appendix A) and the clean separation of the two regimes q \\lessgtr N/s make the contribution sharp and useful for applications that rely on boundary behaviour (comparison, bifurcation, extremal solutions). The technical apparatus—especially the weighted monotonicity via Hardy and the iterative Campanato scheme that upgrades from L^\\infty to any subcritical Hölder exponent—is carefully adapted and of independent interest.","major_comments":[],"minor_comments":[{"comment":"The title page header reads “ON BOUNDAR Y REGULARITY”; correct the spacing.","section":"Title"},{"comment":"Page 2, line after Example 1.1: “and and hence” should be “and hence”.","section":"§1.1"},{"comment":"In the statement of Theorem 1.5 the Hölder exponent \\alpha is left completely undetermined. A brief remark on whether any explicit lower bound (even in terms of N,p,s,q only) can be extracted from the iteration in §4 would help readers who need a concrete modulus.","section":"Theorem 1.5 / §4"},{"comment":"Lemma 2.7 is quoted from earlier work; a one-line indication that the constant C depends only on M,\\beta,\\nu (and not on further geometric data of \\Omega beyond C^{1,1}) would make the interpolation in the proof of Theorem 1.3 for q > N/s fully self-contained.","section":"Lemma 2.7"},{"comment":"Appendix A treats only p = 2. A short sentence clarifying that the same construction is expected to work for p \\neq 2 (or why it does not) would round out the optimality discussion.","section":"Appendix A"},{"comment":"Several constants are labelled C_\\Omega, C_\\alpha, C_\\epsilon without a uniform convention; a single sentence in §1.4 stating the dependence convention would improve readability.","section":"§1.4"}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, carefully written contribution that sits comfortably in the scope of a strong analysis journal. The authors already own much of the bounded-reaction theory; the extension to L^q is genuine and non-trivial. I see no novelty or citation issues."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this closes the gap between the linear L^q theory (Ros-Oton–Serra, Grubb) and the nonlinear bounded-reaction results (their own earlier papers with Squassina): you now get global C^α up to the boundary for f in L^q, q > N/(ps), with the expected split at q = N/s, plus fine boundary regularity of u/d_Ω^s when q > N/s.\n\nWhat is actually new is the simultaneous passage to p ≠ 2 and unbounded reactions. The global statement (Thm 1.3) recovers any α < p'(s − N/(pq)) when N/ps < q ≤ N/s and the sharp Cs when q > N/s; the fine-boundary statement (Thm 1.5) gives some positive Hölder exponent on the quotient. Both come with the natural estimate in terms of ||f||_q^{1/(p−1)}. The proofs stay inside standard nonlocal technique—Campanato oscillations, (s,p)-harmonic extensions, barriers, fractional Hardy, and separate monotonicity for p ≥ 2 and p < 2—and import interior regularity cleanly from Brasco–Lindgren–Schikorra and Garain–Lindgren. The arguments are careful, the limitations (undetermined α in the fine result, failure to hit the endpoint ᾱ, C^{1,1} domain) are stated openly, and the almost-optimality examples are honest.\n\nSoft spots are real but proportionate. The fine-boundary exponent is not explicit, so you cannot yet feed it into sharp bifurcation or free-boundary arguments that need a concrete rate. The global result stops short of C^ᾱ even in regimes where interior theory already reaches it; they conjecture the endpoint but do not prove it. C^{1,1} is essential for the exterior-ball condition and the Hardy lower bound, so Lipschitz or corner domains are out. None of this breaks the theorems as stated.\n\nThis is for people who actually use nonlocal nonlinear regularity in existence, comparison, or bifurcation. The math and citation pattern look solid; no circularity or hidden fitting. I would send it to a serious referee and I would cite the global estimate when I next need boundary Hölder for L^q data. Engage with it.","headline":"Solid, usable extension of boundary Hölder theory for the fractional p-Laplacian from bounded data to L^q reactions, almost optimal and ready to cite.","tokens_in":38378,"tokens_out":577,"would_cite":true,"duration_ms":15508,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","47H11","35A15"],"pacs":[],"model":"grok-4.5","headline":"Solutions of the fractional p-Laplacian with L^q forcing stay Hölder up to the boundary, almost optimally.","keywords":["fractional p-Laplacian","boundary regularity","Hölder continuity","unbounded reactions","fine boundary regularity","nonlocal Dirichlet problem","Campanato estimates"],"falsifier":"Produce a C^{1,1} domain, parameters p,s,q in the stated range, and an L^q reaction whose unique solution fails to be C^α up to the boundary for some α below the claimed threshold, or whose quotient by d_Ω^s fails to be bounded when q > N/s.","tokens_in":38450,"feed_emoji":"∂","tokens_out":923,"duration_ms":18420,"temperature":0.7,"pith_summary":"The paper studies the Dirichlet problem for the s-fractional p-Laplacian with a reaction that is merely integrable to some power q, not necessarily bounded. It proves that the unique weak solution is Hölder continuous all the way to the boundary of a smooth domain, with an explicit exponent that depends on how large q is. When the forcing is summable enough (q larger than N/s), the solution itself is s-Hölder and the quotient of the solution by the s-power of distance to the boundary extends continuously in a Hölder fashion. These statements recover the known linear picture and are shown to be essentially sharp by explicit one-dimensional and higher-dimensional examples. The result matters because global Hölder control and fine boundary behaviour are the ingredients that unlock comparison principles, bifurcation theory, and existence arguments for nonlocal nonlinear equations with rough data.","feed_headline":"Fractional p-Laplace solutions stay Hölder to the boundary","feed_subtitle":"Even with merely L^q forcing, global Hölder and fine boundary regularity hold almost optimally","key_machinery":"Campanato mean-oscillation estimates on (s,p)-harmonic extensions of u inside boundary balls, combined with fractional Hardy inequalities and refined monotonicity of the fractional p-Laplacian (degenerate and singular cases treated separately), which transfer interior oscillation decay and barrier controls up to the boundary.","core_discovery":"For the homogeneous nonlocal Dirichlet problem driven by the s-fractional p-Laplacian with f in L^q, the unique solution u belongs to C^α up to the boundary for every α less than or equal to s that is strictly below p'(s − N/(p q)) when N/(p s) < q ≤ N/s, and for α = s when q > N/s; moreover, when q > N/s the quotient u/d_Ω^s admits a Hölder continuous extension to the closed domain, with a quantitative estimate in terms of the L^q-norm of f.","pith_inferences":["The undetermined Hölder exponent for u/d_Ω^s when q > N/s is likely improvable to the linear value s − N/q by a more precise barrier iteration.","The same oscillation-plus-Hardy scheme should adapt to variable-order or anisotropic fractional p-Laplacians once a corresponding Hardy inequality is available.","Interior C^{ᾱ} theory already known for weaker Lorentz data may combine with the present boundary argument to yield global regularity under those weaker assumptions."],"forward_implications":["Global C^α estimates become available for nonlocal p-Laplace equations with merely integrable right-hand sides, removing the classical L^∞ assumption.","When q > N/s the fine boundary regularity of u/d_Ω^s supplies a continuous fractional normal derivative on ∂Ω.","Existence, comparison, and bifurcation results previously limited to bounded reactions extend immediately to L^q data.","The almost-optimal exponents match the linear fractional Laplacian, confirming that nonlinearity does not destroy the boundary Hölder scale."],"fun_headline_variants":["Fractional p-Laplace solutions Hölder to boundary with L^q data","Boundary Hölder regularity for fractional p-Laplacian, unbounded f","Almost optimal boundary Hölder for nonlocal p-Laplace in L^q","Solutions stay C^α up to boundary for s-fractional p-Laplacian","u/d_Ω^s Hölder extends when q>N/s for fractional p-Laplace"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The domain must have a C^{1,1} boundary so that exterior-ball conditions, distance-function regularity, and the fractional Hardy inequality all hold with uniform constants.","fun_headline_variants_meta":{"raw":{"variants":["Fractional p-Laplace solutions Hölder to boundary with L^q data","Boundary Hölder regularity for fractional p-Laplacian, unbounded f","Almost optimal boundary Hölder for nonlocal p-Laplace in L^q","Solutions stay C^α up to boundary for s-fractional p-Laplacian","u/d_Ω^s Hölder extends when q>N/s for fractional p-Laplace"]},"model":"grok-4.5","effort":"low","cost_usd":0.004836,"raw_usage":{"total_tokens":1353,"prompt_tokens":766,"num_sources_used":0,"completion_tokens":86,"cost_in_usd_ticks":48364000,"prompt_tokens_details":{"text_tokens":766,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":501,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":766,"tokens_out":86,"duration_ms":7728,"temperature":1.0,"reasoning_tokens":501,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T07:23:55.953118+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Produce a C^{1,1} domain, parameters p,s,q in the stated range, and an L^q reaction whose unique solution fails to be C^α up to the boundary for some α below the claimed threshold, or whose quotient by d_Ω^s fails to be bounded when q > N/s.","supporting_citations":[],"review_version":1}