{"id":"38c93c8d-50b1-4a7e-a676-7479f2f69e81","arxiv_id":"2607.16444","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Minimizers of the fractional p-Laplacian one-phase Bernoulli problem exist, are locally Hölder continuous, solve the homogeneous equation in their positivity set, and satisfy the optimal free-boundary growth u(x) ≲ |x−x0|^s.","lead":"This paper proves that minimizers of a free-boundary energy involving the fractional p-Laplacian exist, are locally Hölder continuous, and grow no faster than distance to the free boundary to the power s. The result extends the classical Alt-Caffarelli/Bernoulli regularity theory to a nonlinear, nonlocal setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The τ rescaling in Theorem 1.4 does not normalize sup_{B_1} ũ to 1: the claimed base case (5.9) fails, so the dyadic iteration cannot start as written. The gap is repairable by modifying τ, so the verdict remains CONDITIONAL.","rationale":"The reader's weakest_assumption correctly identifies a genuine flaw in the proof of Theorem 1.4. The displayed τ does not enforce sup_{B_1} ũ ≤ 1; at best it yields sup ≤ 10^s c, and the constant c from (5.1) is not small. Since the entire dyadic iteration starts from (5.9) with k=0, the failure of the base case is load-bearing: without it, Lemma 5.1 cannot be applied to ũ_0, and the optimal growth bound does not follow from the written argument. I agree with the reader's assessment that this is the central gap. The issue is not that the theorem is false; the normalization can be fixed by including sup and Tail and an additional +1 in the denominator of τ, which makes sup_{B_1} ũ ≤ 1, preserves (and even strengthens) the tail bound, and does not affect the smallness of M̃. Thus the appropriate verdict is CONDITIONAL, unchanged from the reader. The minor concerns (the uncited comparison principle for the fractional p-Laplacian and the slightly compressed Harnack step leading to (5.5)) do not affect the main claim; the comparison principle is standard, and the Harnack step is present, albeit stated tersely. Theorems 1.1-1.3 appear well supported. I see no reason to move the verdict.","tokens_in":17703,"tokens_out":11905,"duration_ms":96456,"concrete_test":"Compute sup_{B_1} ũ using the paper's τ with a minimizer-like function having sup_{B_r} u = A and Tail(u;x0,r/2)=0; the result is A/(A/(10^s c)+1). For A = 10^s c, this equals 10^s c / 2, which exceeds 1 for any c satisfying (5.1). This directly disproves the claim (5.9) at k=0. Then verify whether τ' = (A + Tail + 1)^{-1} satisfies the two conditions needed for the iteration: Tail(ũ;0,1/2) ≤ 10^s c and τ'^p r^{sp} M ≤ M0; if both hold, the proof is repairable as suggested.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.4, the rescaling constant is defined as τ = [ (sup_{B_r(x0)} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1 ]^{-1}. The paper asserts sup_{B_1} ũ ≤ 1, but a direct computation gives sup_{B_1} ũ = τ sup_{B_r} u = A/(A/(10^s c)+1), where A = sup_{B_r} u. This quantity is bounded by 10^s c, not by 1. Since c_{n,s,p} is chosen in (5.1) to satisfy (nω_n/sp)^{1/(p-1)} + c 10^{-s/(p-1)} ≤ c, it is necessarily > 0 and in practice > 10^{-s}; hence sup can be arbitrarily large. The 'in particular' average bound (⨍_{B_1} ũ^p)^{1/p} ≤ 1 also fails, so Lemma 5.1 cannot be applied at the first step. Consequently the base case (5.9) for k=0 is unsupported and the dyadic decay/Tail propagation (5.9)-(5.10) does not get off the ground. This is the central estimate of the paper's headline result. The gap appears repairable: choosing τ = (sup_{B_r} u + Tail(u;x0,r/2) + 1)^{-1} (or any normalization with A in the denominator) restores sup_{B_1} ũ ≤ 1, and the tail bound and M̃ ≤ M0 still hold (the tail becomes ≤1, which is even stronger). Thus the theorem is plausible but the proof as written is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the one-phase Alt--Caffarelli (Bernoulli) problem for the fractional $p$-Laplacian, $p\\ge 2$, with a prescribed nonnegative exterior datum and a penalization proportional to the measure of the positivity set. The authors claim existence of minimizers, their nonnegativity and subsolution property (Theorem 1.1), local H\\\"older continuity of minimizers (Theorem 1.2), the Euler--Lagrange equation in the positivity set (Corollary 1.3), and an optimal $r^s$ free-boundary growth estimate (Theorem 1.4). The proofs combine fractional $p$-harmonic replacements, energy-gap estimates, nonlocal tail bounds, a Campanato-type iteration, and a flatness lemma.","tokens_in":18123,"tokens_out":15553,"duration_ms":159561,"significance":"If all results hold, this paper would provide the first systematic free-boundary regularity theory for the Bernoulli problem driven by the fractional $p$-Laplacian, extending the known linear case $p=2$ to the full quasilinear nonlocal range. The proofs of Theorems 1.1--1.3 appear coherent and rely on cited estimates; the paper contains no fitted parameters and produces explicit quantitative bounds. The optimal growth theorem is the headline result and would be a significant contribution. However, as detailed below, the proof of Theorem 1.4 contains a normalization error in the central iterative argument. The gap is localized and appears repairable, so the manuscript is promising, but the main theorem is not established as written.","major_comments":[{"comment":"In the proof of Theorem 1.4, the rescaling constant is defined by τ = ((sup_{B_r(x0)} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1)^{-1}. The paper asserts that sup_{B_1} ũ ≤ 1 and hence (f_{B_1} ũ^p)^{1/p} ≤ 1. A direct computation gives sup_{B_1} ũ = τ sup_{B_r} u = A/(A/(10^s c)+1), where A = sup_{B_r} u; this quantity is < 10^s c, not ≤ 1. Thus the base case (5.9) of the dyadic decay is unsupported, and Lemma 5.1 cannot be applied at the first step. Since (5.9)--(5.10) are the iteration that yields the growth estimate, Theorem 1.4 is not proved as written. The defect appears repairable: for example, a normalization with τ = (A + Tail(u;x0,r/2) + 1)^{-1} makes sup_{B_1} ũ ≤ 1 and Tail(ũ;0,1/2) ≤ 1, but then the constants in Lemma 5.1 and the choice of ε must be rebalanced so that the induction step still produces sup_{B_{1/10}} ũ_k ≤ 10^{-s}. I regard this as a genuine but fixable gap, no","section":"Section 5, proof of Theorem 1.4"}],"minor_comments":[{"comment":"Typo: 'Frational p-harmonic replacement' should read 'Fractional p-harmonic replacement'.","section":"Definition 2.6"},{"comment":"The text refers to 'Theorem 3.2'; the actual reference should be Lemma 3.2 (or Theorem 1.1). Similar cross-reference errors appear elsewhere: Section 4 uses 'Theorem 2.3' for Corollary 2.3 and 'Theorem 4.2' for Lemma 4.2; Section 5 uses 'Theorem 4.1' for Lemma 4.1; and the Euler--Lagrange result is called Corollary 1.3 in the introduction but Theorem 1.3 in Section 4.","section":"Section 3, proof of Corollary 3.3"},{"comment":"The choice of c_{n,s,p} satisfies a lower-bound inequality only if 1 - 10^{-s/(p-1)} > 0; this is true for s>0, but it may help the reader to state explicitly that c_{n,s,p} is chosen sufficiently large.","section":"Section 5, equation (5.1)"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the normalization in the proof of Theorem 1.4. The rest of the paper, in particular Theorems 1.1--1.3, appears sound. I recommend asking the authors to fix the rescaling argument and to correct the internal cross-references. If the normalization gap is resolved, the paper is likely publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is the first systematic treatment of the one-phase Bernoulli problem for the fractional p-Laplacian, and the first three theorems are in good shape. The headline optimal-growth theorem is plausible but its proof has a real normalization gap that will need fixing.\n\nWhat is actually new: existence via the direct method, nonnegativity, the weak subsolution property, local Hölder continuity by harmonic replacement plus Campanato iteration, and the fact that minimizers solve the fractional p-Laplace equation in their positivity set. These results genuinely extend the linear fractional case and the local p-case, and the citation list reflects the relevant prior work. The proofs of Theorems 1.1–1.3 look coherent to me; the tail estimates and the energy-gap lemma are standard and are used correctly.\n\nThe soft spot is Theorem 1.4. The stress-test concern is correct: the rescaling constant τ = [ (sup_{B_r} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1 ]^{-1} does not imply sup_{B_1} ũ ≤ 1. The computation gives sup_{B_1} ũ ≤ 10^s c_{n,s,p}, not 1, and the average bound used in the base case fails for the same reason. The dyadic iteration never gets started as written. This is a genuine gap in a central estimate. It is repairable: taking τ = (sup_{B_r} u + Tail(u;x0,r/2) + 1)^{-1} restores normalization and still gives the tail bound. But the paper needs to fix this before the theorem can be accepted.\n\nTwo lesser points. The comparison principle used in Lemma 5.1 is invoked without a citation; it is standard for the fractional p-Laplacian, but should be stated. The step labeled (5.5) looks like it skips the fact that v(0) ≤ sup_{B_{ι/2}} v; that is immediate, so I do not count it as a real flaw. I also did not find circularities or fitted parameters; the proofs use cited external estimates and the paper's own earlier theorems legitimately.\n\nBottom line: the first half of the paper is solid and worth citing, and the r^s growth rate is very likely correct. This deserves a serious referee even though my own verdict on the current version would be conditional. I would send it to review, and I hope the authors fix the rescaling rather than bury the claim.","headline":"First systematic treatment of the fractional p-Laplacian Bernoulli problem, with solid proofs for existence, Hölder regularity, and the Euler-Lagrange equation in the positivity set; the optimal-growth theorem has a repairable but real rescaling gap.","tokens_in":18631,"tokens_out":6447,"would_cite":true,"duration_ms":48161,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R35","35R11","35J92","35B65","49J40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims optimal order-r^s growth at the free boundary for minimizers of the fractional p-Laplacian Bernoulli problem with p≥2.","keywords":["Alt-Caffarelli problem","fractional p-Laplacian","free boundary","optimal growth","Hölder regularity","nonlocal tail","flatness lemma","Bernoulli problem"],"falsifier":"Directly compute sup_{B_1} ũ for the rescaled function defined in the proof of Theorem 1.4: since ũ(x) = τ u(x0+rx), the supremum over B_1 equals τ sup_{B_r(x0)} u. Inserting τ = [(sup_{B_r} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1]^{-1} gives values up to 10^s c_{n,s,p}, not ≤ 1, unless an additional unstated cancellation occurs; checking whether any hidden estimate forces the supremum below 1 would settle the base case.","tokens_in":17543,"feed_emoji":"📐","tokens_out":3482,"duration_ms":31194,"temperature":0.7,"pith_summary":"The paper studies the one-phase Alt–Caffarelli (Bernoulli) free boundary problem for the fractional p-Laplacian, where a fractional p-energy is penalized by the measure of the positivity set. It establishes existence of minimizers, their nonnegativity and weak subsolution property, and their local Hölder continuity. The central new claim is that at any free boundary point, a nonnegative minimizer grows at most like the distance to the free boundary raised to the power s, matching the sharp order known for the linear fractional Laplacian. This would indicate that the fractional p-Laplacian Bernoulli problem belongs to the same optimal-growth universality class as its linear counterpart.","feed_headline":"Optimal r^s free-boundary growth claimed for fractional p-Laplacian","feed_subtitle":"New iteration extends the sharp linear fractional growth rate to p≥2 via flatness and tail control.","key_machinery":"The proof combines fractional p-harmonic replacements (comparing a minimizer with the solution of the homogeneous equation in a ball with the same exterior values), an energy-gap estimate that controls the L^p difference between the minimizer and its replacement by the volume penalization term, nonlocal tail estimates (including change-of-center and replacement-tail lemmas), and a Campanato-type iteration for local Hölder regularity. The optimal growth proof relies on a flatness lemma for small penalization M and a dyadic iteration that propagates simultaneously a supremum decay and a compatible tail decay across scales.","core_discovery":"For the functional I_M(u) = [u]^p_{W^{s,p}} + M |{u>0} ∩ Ω| with p≥2 and fixed nonnegative exterior data, minimizers exist, are nonnegative, are weak subsolutions of the homogeneous fractional p-Laplace equation, and are locally Hölder continuous. In their positivity set they solve the equation (−Δ)_p^s u = 0. The paper's main result is the optimal free boundary growth estimate: if x0 lies on the free boundary ∂{u>0}, then for small scales r and any x near x0, u(x) ≤ (c/r^s)[Tail(u;x0,r) + (⨍_{B_{2r}} u^p)^{1/p} + 1] |x−x0|^s. This is the same r^s growth rate as in the fractional Laplacian case p=2.","pith_inferences":["The same dyadic-flatness strategy could likely be adapted to prove a matching nondegeneracy estimate, u ≥ c r^s, a natural companion that the paper does not address; together these would pin down the exact growth rate.","The restriction p≥2 appears technical: an analogous r^s growth might be expected for 1<p<2, but the energy-gap monotonicity inequality used here may need a different form in that range.","If the optimal growth theorem is repaired, a natural next step is to study flat free boundary points and prove higher regularity (e.g., C^{1,α}) using the growth estimate as the starting point, following the linear fractional case.","The paper's reliance on the nonlocal tail suggests that the growth estimate is non-local in character; one could test whether the result remains valid with the tail term removed in the bound, which would indicate a stronger local smoothing effect."],"forward_implications":["If the optimal growth estimate holds, minimizers have at most C^s regularity at the free boundary, matching the threshold known for the fractional Laplacian; this is likely the optimal Hölder exponent for this problem.","The subsolution property plus Hölder continuity implies the positivity set is open, and minimizers solve the homogeneous fractional p-Laplace equation where they are positive, allowing standard free-boundary methods to be applied in the positivity set.","The flatness–iteration scheme, once established, provides a template for proving optimal growth for more general nonlinear nonlocal Bernoulli problems with kernels comparable to |y|^{-n-sp}.","The energy-gap estimate gives a quantitative way to transfer regularity from fractional p-harmonic functions to minimizers, useful beyond the Bernoulli context for other free boundary problems with nonlocal nonlinear diffusion."],"fun_headline_variants":["Sharp r^s growth for fractional p-Laplacian free boundary","Optimal growth proved for p≥2 fractional Bernoulli","Fractional p-Laplacian: Hölder regularity and optimal growth","New iteration in p≥2: optimal free-boundary growth","Existence, regularity, sharp growth for p≥2 fractional"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of the optimal growth estimate depends on a rescaling constant τ chosen so that the rescaled function satisfies sup_{B_1} ũ ≤ 1; a direct computation shows sup_{B_1} ũ = τ sup_{B_r} u can be as large as 10^s c_{n,s,p}, generally exceeding 1, so the base case of the dyadic decay iteration is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Sharp r^s growth for fractional p-Laplacian free boundary","Optimal growth proved for p≥2 fractional Bernoulli","Fractional p-Laplacian: Hölder regularity and optimal growth","New iteration in p≥2: optimal free-boundary growth","Existence, regularity, sharp growth for p≥2 fractional"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000367,"raw_usage":{"total_tokens":1804,"prompt_tokens":739,"completion_tokens":1065,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":979}},"tokens_in":483,"tokens_out":1065,"duration_ms":9187,"temperature":1.0,"reasoning_tokens":979,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T20:57:54.116177+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly compute sup_{B_1} ũ for the rescaled function defined in the proof of Theorem 1.4: since ũ(x) = τ u(x0+rx), the supremum over B_1 equals τ sup_{B_r(x0)} u. Inserting τ = [(sup_{B_r} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1]^{-1} gives values up to 10^s c_{n,s,p}, not ≤ 1, unless an additional unstated cancellation occurs; checking whether any hidden estimate forces the supremum below 1 would settle the base case.","supporting_citations":[],"review_version":1}