{"id":"bd93dee0-3490-4f5c-bfd9-e4f7de3297bc","arxiv_id":"2507.20581","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A C^{2,alpha} free boundary in the nonlocal one-phase problem is proved to be C^infinity for general integro-differential operators of order 2s.","lead":"This mathematics paper proves that if the free boundary in a nonlocal Bernoulli-type problem is smooth to second order, then it is actually infinitely smooth. This settles a long-open regularity question even for the fractional Laplacian with order other than 1/2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.7 needs J∈C^{1,δ} with δ>2s−1, but C^{2,α} with α≤2s−1 only gives J∈C^{1,α}; the bootstrap in Theorem 1.1 misses this case.","rationale":"The reader identified the Hopf-type nondegeneracy (6.2) as the weakest assumption. I do not think that is the main problem: Lemma 2.3 supplies ∂_n v ≥ c d^{s−1}, and because d^{s−1} blows up near the boundary, (6.2) can be verified in a small ball after shrinking. The gap I see is a regularity-threshold mismatch in the integration-by-parts machinery that converts the quotient equation into the boundary weak form. Lemma 3.7 is stated with J∈C^{1,δ}, δ>max{0,2s−1}; the available J is only C^{1,α} when ∂Ω∈C^{2,α}. For s>1/2 and α≤2s−1 this is not minor: the same δ appears in a one-dimensional integral that must converge, and below δ=2s−1 it diverges. The theorem states all α>0, so the proof as written does not cover this regime. This does not make the result false; the missing case may be treatable by exploiting the special structure of J or by a different integration by parts, and the paper's new identities, weighted Liouville theorems, and boundary estimates are substantial and independently valuable. Therefore the verdict should be CONDITIONAL rather than an unconditional accept or a rejection: the central claim is plausible but the proof needs an additional argument or a stated restriction on α.","tokens_in":90930,"tokens_out":38163,"duration_ms":377085,"concrete_test":"Take s=0.8, α=0.1, k=2, and let ∂Ω∈C^{2,0.1} with v/d^s=A(ν). From Lemma 6.1(i), the available regularity of J=d^{1−s}∂_n u∘Φ |detDΦ| is C^{1,0.1}. Try to reproduce the proof of Lemma 3.7 with δ=0.1: the term J_ε^(2) (or its flattened analogue) contains ∫_0^ε r^{s−1}∫_ε^∞ t^{s−1}|r−t|^{−2s+δ} dt dr with s=0.8 and δ=0.1; the exponent is −1.5 and the integral diverges as ε→0. Hence Lemma 3.7 cannot be invoked in this case. A decisive check would be to replace the hypothesis J∈C^{1,δ} by the product structure J=sA(ν)J_0+d(...) and prove the same convergence with δ=α; if this cannot be done, Theorem 1.1 must be restricted to α>max{0,2s−1} or supplied with an additional initial regularity-improvement argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 6.7 and Theorem 1.1 start with ∂Ω∈C^{2,α} for arbitrary α>0. To run the bootstrap, Corollary 6.2(iv)/6.6(iii) must convert the equation for the incremental quotient W into the weak form used by Proposition 5.1. This conversion is Lemma 3.7, whose hypotheses include J∈C^{1,δ}_c for some δ∈(max{0,2s−1},s). In the application, J=d^{1−s}∂_n u∘Φ |detDΦ|. Lemma 6.1(i) (via Lemma 2.2) only yields J∈C^{k−1,α}; for the initial step k=2 this is C^{1,α}. When s>1/2 and α≤2s−1, no admissible δ exists. The proof of Lemma 3.7 genuinely uses δ>2s−1, for example in the treatment of J_ε^(2): after Lemma 7.3 one obtains powers |x_n−y_n|^{−2s+δ}, and the double integral ∫_0^ε r^{s−1}∫_ε^∞ t^{s−1}|r−t|^{−2s+δ} dt dr converges only if δ>2s−1. Thus the weak formulation and the subsequent C^γ estimate are not justified in the regime α≤max{0,2s−1} that the theorem statement includes. The paper never states an α>max{0,2s−1} restriction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops new higher-regularity tools for nonlocal free boundary problems. The main result (Theorem 1.1) states that for a general nonlocal operator L of order 2s with smooth kernel on the sphere, a C^{2,α} free boundary (for any α>0) of a solution to the nonlocal one-phase problem is actually C^∞. A parallel result (Theorem 1.2) is proved for overdetermined problems with smooth boundary data, and a new proof is given for the nonlocal obstacle problem (Theorem 1.3). The technical core consists of new integration by parts formulas for weighted nonlocal energies (Lemma 3.1 and Lemma 3.7), weighted Liouville theorems (Proposition 4.1), boundary Hölder estimates for equations with local Neumann-type conditions (Proposition 4.3 and Proposition 5.1), and a bootstrap argument on the quotient ∂_i u/∂_n u.","tokens_in":91249,"tokens_out":9422,"duration_ms":89717,"significance":"If the proof is correct as stated, the paper solves a long-standing open problem: the C^∞ regularity of free boundaries in the nonlocal one-phase problem was known only for s=1/2, so Theorem 1.1 is a major advance even for the fractional Laplacian. The paper also introduces genuinely new integration by parts identities and boundary Hölder estimates for nonlocal equations with singular weights, which are likely to be of independent interest. The approach is robust and unified, covering the one-phase, overdetermined, and obstacle problems. The proofs are detailed and mostly self-contained, with standard technical lemmas deferred to well-known references.","major_comments":[{"comment":"Lemma 3.7 requires J∈C^{1,δ}_c for some δ∈(max{0,2s−1},s). In the application to the one-phase and overdetermined problems, J(x)=(d^{1−s}∂_n u∘Φ)(x)|detDΦ(x)|. Lemma 6.1(i), via Lemma 2.2, gives only d^{1−s}∇u∈C^{k−1,α}; for the initial step k=2 this yields J∈C^{1,α}. If s>1/2 and α≤2s−1, no admissible δ exists. The proof of Lemma 3.7 genuinely uses δ>2s−1: in the treatment of the term J_ε^(2) (after (3.19)–(3.20)), Lemma 7.5 is applied to the double integral ∫_0^ε r^{s−1}∫_ε^∞ t^{s−1}|r−t|^{-2s+δ} dt dr, which converges only when δ>2s−1. Consequently the weak formulation for the incremental quotient W^{(h)} in Corollary 6.2(iv) and Corollary 6.6(iii) is not justified in this regime, and Proposition 5.1 cannot be invoked as in Propositions 6.3 and 6.7. Since Theorem 1.1 is stated for arbitrary α>0, the proof does not cover α≤max{0,2s−1} when this quantity is positive. The same gap affects Theorem 1.2, because the same J-regularity hypothesis is used. The manuscript should either restrict the statements to α>max{0,2s−1} or provide an integration by parts result valid when J∈C^{1,α} only with α>0.","section":"§3.2 (Lemma 3.7) and §6.1 (Corollary 6.2(iv), Corollary 6.6(iii))"}],"minor_comments":[{"comment":"Theorem 1.1 states that the result holds for 'any solution' of (1.3), but the proof in §6.1.2 is carried out for minimizers in the sense of Definition 2.4 (see Proposition 6.7 and Lemma 2.5). The manuscript should state explicitly the class of solutions covered, or explain why every solution of (1.3) is a minimizer.","section":"§1.3, Theorem 1.1 and §6.1.2, Proposition 6.7"},{"comment":"The remark after Corollary 6.2(iv) says that when α<max{0,2s−1,1/2} one may apply the result with k̄=k−1 and a larger ᾱ. This does not resolve the J-regularity issue raised in the major comment, because Lemma 3.7 still requires J∈C^{1,δ} with δ>2s−1. The remark should be reconciled with the hypotheses of Lemma 3.7.","section":"§6.1.1, Corollary 6.2, remark after (iv)"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are likely true and the techniques are significant, but the gap around Lemma 3.7 affects the stated range of α in Theorems 1.1 and 1.2. The issue is concentrated in one lemma and its application, so it should be fixable by either strengthening that lemma for lower-regularity J or by adjusting the statements and their hypotheses. I would not reject the paper, but the current version overclaims the range of α."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is the real thing. It proves that C^{2,α} free boundaries are C^∞ for the nonlocal one-phase problem for general kernels and every s, which was open for the fractional Laplacian except s=1/2. Theorem 1.2 on overdetermined problems is also new, and Theorem 1.3 gives a genuinely different proof for the obstacle problem. The machinery is substantial: new integration by parts formulas for weighted nonlocal energies, boundary Hölder estimates for equations with local Neumann conditions, and a weighted Liouville theorem. I read the proof structure carefully and the main line holds together. The reliance on the authors' own prior work for the boundary condition A(ν) is legitimate; that is a separate established result, not a circular step.\n\nThe soft spot is in the hypotheses of Lemma 3.7. The lemma requires J ∈ C^{1,δ} with δ > max{0,2s−1}, but the application only provides J ∈ C^{1,α} from Lemma 6.1. When α is small (which happens when s > 1/2 and α ≤ 2s−1), no admissible δ exists, and Corollaries 6.2 and 6.6 invoke Lemma 3.7 in a regime where its stated assumptions are not met. The stress-test note flagged this, but its specific justification is wrong: the double integral in question converges for δ > 2s−2, not δ > 2s−1, so the proof of Lemma 3.7 actually goes through for any δ > 0. The gap is in the lemma statement, not the underlying mathematics. The fix is simple—weaken the hypothesis to δ ∈ (0,s) and verify the proof, which already works. That is a presentation repair, not a fatal flaw.\n\nThe paper is 85 pages and some standard De Giorgi steps are left to the reader, so verification is heavy. But I found no load-bearing error. This is for anyone working in free boundaries or nonlocal elliptic regularity, and it will be cited. It deserves a serious referee; I would send it to review and ask for the lemma repair and a remark clarifying the α range in the bootstrap. Accept after minor revision.","headline":"A major open problem in nonlocal free boundaries is settled, with one presentation gap in the regularity assumptions of the key integration-by-parts lemma that should be fixed before publication.","tokens_in":91816,"tokens_out":18872,"would_cite":true,"duration_ms":163511,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R35","47G20","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for general nonlocal operators of order $2s$, any $C^{2,\\alpha}$ free boundary in the one-phase Bernoulli problem is automatically $C^{\\infty}$, new even for the fractional Laplacian when $s\\neq 1/2$.","keywords":["nonlocal free boundary problems","one-phase Bernoulli problem","fractional Laplacian","integro-differential operators","higher regularity","obstacle problem","oblique boundary conditions","weighted nonlocal equations"],"falsifier":"Solve the equation $Lv=0$ in $\\Omega$, $v=0$ outside, with boundary scaling $v/d_{\\Omega}^s$ normalized to $A(\\nu)$, on a domain whose boundary is deliberately $C^{2,\\alpha}$ but not $C^3$ at one point, and compute the tangential quotient $\\partial_i v/\\partial_n v$ along points approaching that point. The theorem predicts this quotient and all its tangential incremental quotients remain bounded and converge to a smooth trace; if the quotient develops a singularity or the incremental quotients fail to converge for some $k$, the claim is false. A simpler diagnostic is the liminf of $r^{1-s}\\partial_\\nu v$ at regular boundary points: finding a point where this liminf is $0$ while $v$ is not identically zero would break the argument and, if the boundary is not $C^{\\infty}$ there, falsify Theorem 1.1.","tokens_in":90729,"feed_emoji":"📐","tokens_out":11798,"duration_ms":111667,"temperature":0.7,"pith_summary":"This paper proves a regularity bootstrapping statement for nonlocal free boundary problems: if a free boundary is already known to be $C^{2,\\alpha}$, then it is in fact $C^{\\infty}$. The main case is the nonlocal one-phase (Bernoulli) problem for a general integro-differential operator of order $2s$ with a smooth kernel on the sphere, and the conclusion is new even for the fractional Laplacian when $s\\neq 1/2$. The same method shows that in overdetermined problems, smooth boundary data forces the boundary to be smooth, and it yields a fresh proof, not based on higher-order boundary Harnack inequalities, that regular free boundaries in the nonlocal obstacle problem are $C^{\\infty}$. The engine is a quotient of derivatives of the solution that satisfies a weighted nonlocal equation with a local Neumann-type boundary condition, together with new integration-by-parts formulas and boundary H\\\"older estimates for such equations.","feed_headline":"C^{2,α} free boundaries become C^∞ in nonlocal problems","feed_subtitle":"A quotient-of-derivatives argument with new integration by parts extends the result beyond the fractional Laplacian.","key_machinery":"The load-bearing object is the quotient $w=\\kappa_2\\,\\partial_i u/\\partial_n u$ of the truncated solution $u$, together with the weighted nonlocal equation it satisfies: $\\int_{\\Omega}(w(x)-w(y))\\,\\partial_n v(x)\\,\\partial_n v(y)\\,K(x-y)\\,dy=0$ in $\\Omega$, with a local Neumann-type boundary condition on $\\partial\\Omega$. Near the boundary the weights behave like $d_{\\Omega}^{s-1}(x)\\,d_{\\Omega}^{s-1}(y)$, and the proof reduces the matter to a model half-space problem with weight $(x_n)_+^{s-1}(y_n)_+^{s-1}$ and Neumann condition $\\partial_n w=0$ on $\\{x_n=0\\}$. Three new tools carry the argument: integration-by-parts formulas (Lemma 3.7) that turn the weighted nonlocal energy into a bulk term plus a boundary term involving the oblique derivative $\\Theta_{K,\\Omega}\\cdot\\nabla w$; a De Giorgi iteration with weighted Poincar\\'e--Sobolev inequalities yielding boundary H\\\"older estimates and a weighted Liouville theorem (only constants solve the model problem); and a Liouville-based compactness argument producing the a priori boundary H\\\"older estimate of Proposition 5.1. For the one-phase problem, the kernel's first-moment direction $\\theta_K=2c_s\\int_{\\mathbb{R}^{n-1}}(h',1)K((h',1))\\,dh'$ is handled by a shear transformation (Lemma 3.13) that converts the oblique boundary condition into a normal one for a modified kernel, and the identity $A(\\nu)=c_{n,s}(\\int_{S^{n-1}}K(\\theta)|\\theta\\cdot\\nu|^{2s}d\\theta)^{-1/2}$ is used to show that the boundary condition for $w$ is exactly $\\partial_{\\theta(x)}w=0$ along $\\theta(x)\\parallel\\nabla A(\\nu(x))$.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.1: let $L$ be an operator of the form (1.1)--(1.2) whose kernel $K$, restricted to the unit sphere, is $C^{\\infty}$, and let $v\\in L^{\\infty}(\\mathbb{R}^n)$ be any solution of the nonlocal one-phase free boundary problem (1.3). If $\\partial\\Omega\\cap B_1$ is in $C^{2,\\alpha}$, then $\\partial\\Omega\\cap B_{1/2}$ is in $C^{\\infty}$. The mechanism is the tangential quotient $w=\\kappa_2\\,\\partial_i u/\\partial_n u$: after flattening the boundary, $w$ satisfies a weighted nonlocal equation with a local Neumann or oblique boundary condition, and the proof shows that $w$ gains H\\\"older regularity of order $\\gamma$ at the boundary for every $\\gamma\\in(\\max\\{1/2,2s-1\\},1)$. Iterating this gain through tangential incremental quotients lifts the boundary regularity from $C^{k,\\alpha}$ to $C^{k+\\alpha+\\gamma}$, and repeating the step reaches $C^{\\infty}$. The analogous Theorem 1.2 treats overdetermined problems with smooth boundary datum $h$, and Theorem 1.3 gives the same conclusion $C^{1,\\alpha}\\Rightarrow C^{\\infty}$ for regular free boundaries of the nonlocal obstacle problem.","pith_inferences":["Because the $C^{\\gamma}$ gain in Proposition 5.1 depends only on dimension, ellipticity bounds, and the H\\\"older norm of the kernel, the $C^{2,\\alpha}\\Rightarrow C^{\\infty}$ conclusion should be stable under small $C^{\\infty}$ perturbations of the kernel, with uniform constants.","The shear/oblique-to-normal reduction isolates the kernel through the single vector $\\theta_K$; this suggests anisotropic kernels will produce free boundaries that are smooth but whose local coordinate system is tilted by $\\theta_K$, an anisotropy effect that could be detected by computing $\\theta_K$ for a specific non-symmetric kernel.","The result implicitly supports a gain-of-one-derivative-per-iteration heuristic: each application of Proposition 5.1 upgrades boundary regularity by a fixed H\\\"older amount, so the number of iterations needed to reach $C^{\\infty}$ is controlled by the initial $\\alpha$ and by $s$.","For the obstacle problem, the new proof suggests higher-order boundary Harnack machinery is not essential for smoothness, opening the door to treating nonlinear nonlocal obstacle-type equations by the same quotient and Neumann-weight method."],"forward_implications":["Completing the $C^{2,\\alpha}\\Rightarrow C^{\\infty}$ step, the only missing piece in the nonlocal one-phase programme is upgrading flat boundaries to $C^{2,\\alpha}$; the authors state they will supply it in a future paper, at which point the full chain flat $\\Rightarrow C^{1,\\alpha}\\Rightarrow C^{2,\\alpha}\\Rightarrow C^{\\infty}$ holds.","For the nonlocal obstacle problem, the smoothness of regular free boundaries now has a proof independent of higher-order boundary Harnack inequalities.","For overdetermined problems, the result is a reverse regularity statement: smoothness of $u/d_{\\Omega}^s$ on the boundary implies smoothness of the boundary itself, answering the nonlocal analogue of the classical Poisson-kernel question.","The new boundary H\\\"older estimates and integration-by-parts formulas apply to the whole range of weights $(x_n)_+^{\\beta-1}$ with $\\beta\\in[s,1+s]$, and the paper notes the intermediate range is relevant to the nonlocal Alt--Phillips problem, so the same machinery is likely reusable there.","As a corollary, the theorem converts the regularity question for the one-phase problem into the flatness/$C^{1,\\alpha}$ step: any improvement of flatness up to $C^{2,\\alpha}$ automatically yields infinite smoothness."],"supporting_citations":[{"why":"Supplies the boundary condition $v/d_{\\Omega}^s=A(\\nu)$, the minimizer properties, and the $C^{1,\\alpha}$ flatness ingredients that Theorem 1.1 builds on.","marker":"[RoWe24b]"},{"why":"Provides the starting regularity for nonlocal equations with local Neumann boundary conditions and the kernel estimates used in Lemma 6.1.","marker":"[RoWe24a]"},{"why":"Supplies the interior Schauder and Cordes-Nirenberg estimates with singular kernels used in the blow-up arguments of Lemma 5.3 and Lemma 5.4.","marker":"[FeRo24b]"},{"why":"Provides the flattening diffeomorphism, boundary regularity estimates, and the obstacle-problem higher regularity result that the new proof replaces.","marker":"[AbRo20]"},{"why":"Contains the Green's identity precursor that the new integration-by-parts formulas generalize.","marker":"[Gru20]"},{"why":"Supplies the one-dimensional Liouville theorem for the fractional Laplacian used in the $\\beta=s$ case of the weighted Liouville theorem.","marker":"[RoSe16]"},{"why":"Supplies the weighted Poincar\\'e and Poincar\\'e-Sobolev inequalities and De Giorgi iteration framework adapted in Section 4.","marker":"[BDOR24]"},{"why":"Supplies the nonlocal Caccioppoli inequality pattern used in Lemma 4.7.","marker":"[DKP16]"}],"fun_headline_variants":["Nonlocal free boundaries: C^{2,α} implies C^∞","From C^{2,α} to C^∞: regularity in nonlocal Bernoulli","Infinite smoothness from C^{2,α} in nonlocal free boundaries","C^{2,α} free boundaries are C^∞ for nonlocal operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof requires that near the boundary point the solution grows out of the boundary at a non-degenerate rate: its normal derivative must stay bounded below by a positive constant times (distance to the boundary)$^{s-1}$, and the normal derivative of the distance function must also stay positive. If that lower bound vanishes at a boundary point, the quotient $w$ is not defined and the bootstrap cannot start.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal free boundaries: C^{2,α} implies C^∞","From C^{2,α} to C^∞: regularity in nonlocal Bernoulli","Infinite smoothness from C^{2,α} in nonlocal free boundaries","C^{2,α} free boundaries are C^∞ for nonlocal operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001115,"raw_usage":{"total_tokens":4678,"prompt_tokens":1016,"completion_tokens":3662,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":3575}},"tokens_in":632,"tokens_out":3662,"duration_ms":26265,"temperature":1.0,"reasoning_tokens":3575,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:40:30.870841+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the equation $Lv=0$ in $\\Omega$, $v=0$ outside, with boundary scaling $v/d_{\\Omega}^s$ normalized to $A(\\nu)$, on a domain whose boundary is deliberately $C^{2,\\alpha}$ but not $C^3$ at one point, and compute the tangential quotient $\\partial_i v/\\partial_n v$ along points approaching that point. The theorem predicts this quotient and all its tangential incremental quotients remain bounded and converge to a smooth trace; if the quotient develops a singularity or the incremental quotients fail to converge for some $k$, the claim is false. A simpler diagnostic is the liminf of $r^{1-s}\\partial_\\nu v$ at regular boundary points: finding a point where this liminf is $0$ while $v$ is not identically zero would break the argument and, if the boundary is not $C^{\\infty}$ there, falsify Theorem 1.1.","supporting_citations":[],"review_version":2}