{"id":"0e7bcb77-3bfa-4a19-a786-61fc685b0f7b","arxiv_id":"2507.10336","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Alt-Phillips free boundary problem, the paper proves smoothness of regular free boundaries for all exponents, derives a stability inequality for negative exponents, and rules out nontrivial axially symmetric stable cones in dimensions d≤6 (and d=7 for γ below about -0.717).","lead":"This paper proves that regular free boundaries in the Alt-Phillips problem are smooth for every exponent in (-2,2), closing the previously open negative-exponent case, and derives a new stability condition for minimizing cones in that regime. It also shows that axially symmetric stable cones are trivial in low dimensions and connects the stability criterion to minimal surfaces as the exponent approaches -2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's reduction of local minimizers to regular solutions rests on Lemma 2.3, whose proof is a compressed upgrade from C^{1,δ} to C^{1,α}, α > -s, without full details; this is the least secure link.","rationale":"The reader identified Lemma 2.3 as the weakest assumption, and I agree. My independent check of the rest of the argument found no clearer or more serious flaw: the hodograph transformation is standard, the bootstrap for the degenerate quasilinear equation in Proposition 2.8 is plausible and internally consistent, the stability computation in Section 3 uses smoothness that Proposition 2.2 provides, and the Hardy-type arguments in Section 4 appear algebraically correct. The unproven uniform regularity assumption (5.6) in Section 5 is a secondary gap that the paper itself acknowledges, and it does not affect Theorem 1.1. Thus the same conditionality the reader already placed is appropriate: the main theorem is convincing but not fully verified at the point where local minimizers are matched to the regular-solution framework. If Lemma 2.3 were supplied with complete details, the central claim of the paper would go through as stated. I do not see grounds to strengthen the verdict to accept, nor to reject.","tokens_in":31542,"tokens_out":28801,"duration_ms":318961,"concrete_test":"Independently re-derive Lemma 2.3 from [20, Prop. 7.2] and [45, Theorem 1.1], tracking the Hölder exponent in the linearized improvement-of-flatness. Specifically, check whether the estimate for solutions of div(x_d^s ∇v) = 0 with the Neumann condition lim_{x_d→0} x_d^s ∇v · e_d = 0 yields C^{1,α} regularity up to the boundary for every α < 1, with constants that allow an iteration reaching the required α > -s. If the linearized theory caps at some β ≤ -s, then Theorem 1.1 fails for exponents with -s > β; alternatively, run the iteration on the explicit model w(t) = t + c t^{2-s} to see whether the free-boundary condition and the claimed decay ρ^{1+α} are compatible for all α > -s.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 1.1, is proved by first establishing Proposition 2.2 for regular solutions and then invoking Lemma 2.3 to assert that every local minimizer is a regular solution near a regular free boundary point. Lemma 2.3 is therefore load-bearing: if the upgrade from the known C^{1,δ} regularity of [20, Theorem 2.3] to C^{1,α} with α > -s fails, the hodograph and Schauder machinery in Proposition 2.2 never starts for local minimizers. The proof of Lemma 2.3 is only a sketch: it cites [20, Prop. 7.2] and the smoothness of the linearized problem from [45, Theorem 1.1], then asserts that 'for every α ∈ (0,1)' an improvement-of-flatness estimate holds with decay ρ^{1+α}, and concludes by a 'standard iteration'. What is not shown is that the radius ρ in the improvement step can be chosen uniformly in ε for the particular α > -s required by Definition 2.1, nor that the weighted free-boundary condition w^s(|∇w|²−1) → 0 is preserved through the iteration with the stated pointwise limit. For γ ∈ (-2,0), Definition 2.1 demands α > -s, which can be arbitrarily close to 1 as γ → -2; if the linearized estimates for the operator div(x_d^s ∇v) = 0 only yield a fixed Hölder exponent bounded by -s, the iteration would cap out below the required threshold and Lemma 2.3 would fail exactly in the regime the paper targets. This is not an internal contradiction, but it is the weakest step in the proof of the main theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the one-phase Alt–Phillips functional J_γ for γ∈(-2,2), with emphasis on the negative-exponent range γ∈(-2,0). For regular free boundary points it proves C^∞ smoothness of the free boundary and of u/dist(·,∂Ω_u)^β (Theorem 1.1), by a hodograph transform that reduces the problem to a degenerate quasilinear PDE div(x_d^s DF(∇h))=0 with a Neumann boundary condition, for which the authors establish a Schauder theory (Theorem 1.2). A preliminary step, Lemma 2.3, asserts that local minimizers are regular solutions near regular points. The paper then computes the second inner variation of E_s, derives a Sternberg–Zumbrun-type stability inequality (Theorem 1.3), uses it to prove that axially symmetric β-homogeneous minimizers are one-dimensional in low dimensions (Theorem 1.4), and reformulates the stability inequality as an eigenvalue bound on S^{d-1} (Proposition 5.1). A final section discusses the limit γ→-2 and claims convergence of the stability criterion to the minimal-surface criterion, under an explicit unproved uniform C^{2,α} compactness assumption (5.6).","tokens_in":31828,"tokens_out":12973,"duration_ms":133645,"significance":"If the results are correct, Theorem 1.1 completes the higher-regularity program for the Alt–Phillips problem by covering negative exponents, and the unified hodograph/Schauder approach is a genuine methodological advance. The Schauder estimates for degenerate quasilinear operators in Theorem 1.2 are of independent interest and are derived with parameter-free arguments and no fitted constants. The stability inequality and the axial-symmetry rigidity are new in the negative-exponent regime and establish a bridge to the Stability theory of minimal surfaces. The main caveat is that the advertised recovery of the minimal-surface stability criterion in the singular limit γ→-2 is conditional on assumption (5.6), which the authors explicitly state is not proved; the text is honest about this, but the abstract is not.","major_comments":[{"comment":"The proof of Lemma 2.3 is a two-paragraph sketch that is load-bearing for Theorems 1.1, 3.3, and 1.3. It asserts an improvement-of-flatness estimate valid for every α∈(0,1) with a radius ρ uniform in the flatness parameter ε, citing [20, Prop. 7.2] and [45, Thm. 1.1], then concludes by a 'standard iteration'. The manuscript should provide the full iteration: (i) justify that the radius ρ can be chosen independently of ε and uniformly for the required α > -s, in particular for α arbitrarily close to 1 as γ→-2; (ii) show explicitly that the pointwise weighted free-boundary condition lim_{t→0} w^s(|∇w|^2-1)=0 is preserved through the iteration; (iii) either reproduce the argument or give a precise reference for the claimed C^{1,α} upgrade with α > -s. Without this, Proposition 2.2 applies only to regular solutions, and the reduction from local minimizers to that class is incomplete.","section":"Section 2, Lemma 2.3"},{"comment":"The abstract states that the variational criterion 'recovers the one for minimal surfaces in the singular limit as γ→-2', but the only result in this direction, Section 5.2, is conditional on assumption (5.6), which the authors explicitly do not prove ('The proof of assumption (5.6) is rather involved and would require a more detailed refinement of our regularity theory, which goes beyond the scope of this paper'). The abstract and introduction should either present this as a conditional result or incorporate assumption (5.6) into the advertised statement.","section":"Abstract and Section 5.2"},{"comment":"The statement says that stability in R^d is equivalent to the eigenvalue lower bound (5.2), but the proof only establishes equivalence with the scalar inequality (1.4) for test functions of the separated form f(r,θ)=g(r)φ(θ). The converse direction, that the scalar inequality (1.4) implies the full inner-variation stability of Definition 3.1, is not shown. Please either restrict Proposition 5.1 to the scalar stability condition or supply the missing implication.","section":"Section 5.1, Proposition 5.1"}],"minor_comments":[{"comment":"The sentence 'Such method provide a unified proof' should be corrected to 'This method provides a unified proof' or 'Such a method provides a unified proof'.","section":"Abstract"},{"comment":"The phrase 'ζε isfirstchosenasanaxiallysymmetriccut-offfunction, andsubsequentlyreplaced by a radial one' is missing spaces due to LaTeX formatting; it should read 'ζ_ε is first chosen as an axially symmetric cut-off function, and subsequently replaced by a radial one'.","section":"Section 4, proof of Proposition 4.1, Step 3"},{"comment":"The threshold 'γ <10 −8√5 11 ≈ −0.7171' is ambiguous; it should be written as γ < (10 − 8√5)/11 ≈ −0.7171.","section":"Theorem 1.4"},{"comment":"The notation 'M := lim_{k→∞} ∂Σ_{w_k}' uses a limit of sets without specifying the notion of convergence; the subsequent argument suggests Hausdorff or varifold convergence, so the mode of convergence should be stated.","section":"Section 5.2"},{"comment":"Reference [39] (Pacati, Tortone, Velichkov) appears in the reference list but is not cited in the text; either add a citation or remove it.","section":"References"},{"comment":"The proposition is stated in terms of w but says 'u is stable in R^d if and only if λ_s(Σ_w) ≥ ...'; since u and w are equivalent, the statement should clarify that stability of u is equivalent to the eigenvalue bound for w, to avoid confusion.","section":"Section 5.1, Proposition 5.1"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with a solid new Schauder theory and interesting applications. The main issues are the compressed proof of Lemma 2.3 and the abstract's overclaim regarding the γ→-2 recovery, which is conditional on assumption (5.6). The reliance on the authors' own prior linear theory [47] is appropriate and does not amount to circularity. I would support publication after the flagged points are addressed, either by adding the missing details for Lemma 2.3 or by clearly stating the conditional nature of the asymptotic result in the abstract and introduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious piece of work. It proves C∞ smoothness of the regular free boundary for negative Alt-Phillips exponents, which was open, and gives a unified proof for the full range γ∈(−2,2). The quasilinear Schauder theory in Theorem 1.2 is genuinely new and useful on its own; the stability inequality and the axially symmetric cone rigidity are solid contributions. The hodograph transformation and the bootstrap are detailed and internally coherent. I checked the main identities and they hold up. The paper is honest about what is proved and what is not, and the reliance on the authors' prior linear theory in [47] is legitimate — that theory is published and parameter-free, and the quasilinear extension is a real step beyond it.\n\nThe weak link is Lemma 2.3. It is load-bearing: Theorem 1.1 for local minimizers rests on it, and the proof is a sketch. The stress-test concern has teeth. The improvement-of-flatness step needs a radius that is uniform for the specific α > −s required by Definition 2.1, and neither the paper nor the cited [20, Prop. 7.2] shows that uniformity. Near γ→−2, α can be arbitrarily close to 1, and if the linearized estimates only give a fixed exponent below that threshold, the iteration caps out. This is not an internal contradiction, and the gap is probably fillable, but the paper should either supply the full iteration or state Theorem 1.1 for regular solutions and only conjecture the minimizer step.\n\nSecond issue: the abstract says the stability criterion recovers the minimal-surface criterion as γ→−2, but assumption (5.6) — uniform C^{2,α} regularity — is explicitly unproven and the paper says so in Section 5. That claim should be qualified in the abstract.\n\nWho this is for: anyone working on free boundary regularity or degenerate PDEs. Theorem 1.2 alone is worth the read, and the stability computation in Section 3 is careful and useful. I would send this to a serious referee. The right expectation is a major revision that closes or clearly restricts Lemma 2.3 and tones down the abstract. I'd cite it for the Schauder theorem even if the minimizer-to-regular reduction stays open.","headline":"Serious paper that proves a long-open smoothness result, with a real but likely fixable gap in the reduction from minimizers to regular solutions and an overclaim in the abstract.","tokens_in":32416,"tokens_out":1693,"would_cite":true,"duration_ms":20703,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R35","35B65","35J61","35B07","49Q05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that regular free-boundary points in the Alt-Phillips problem are smooth for every exponent $\\gamma\\in(-2,2)$, and derives a stability inequality that makes axially symmetric cones one-dimensional in low dimensions.","keywords":["Alt-Phillips","free boundary","higher regularity","hodograph transform","degenerate quasilinear equations","Schauder estimates","stable solutions","stable minimal cones"],"falsifier":"A concrete check would be to construct a solution of $\\operatorname{div}(x_d^s DF(\\nabla v))=0$ with $F$ uniformly convex and $F\\in C^{3,\\alpha}$ that is $C^{1,\\alpha}$ but not $C^{2,\\alpha}$ in $B^+_{1/2}$; such a counterexample would falsify Theorem 1.2 and with it the smoothness claim. On the stability side, evaluating inequality (4.2) on any explicit nontrivial axially symmetric 1-homogeneous cone in dimension 6 would settle Theorem 1.4.","tokens_in":31222,"feed_emoji":"📐","tokens_out":13363,"duration_ms":128476,"temperature":0.7,"pith_summary":"At regular free-boundary points, the Alt-Phillips interface is smooth for every exponent $\\gamma\\in(-2,2)$; the paper closes the negative-exponent gap and gives one unified proof. This matters because the Alt-Phillips functional interpolates between the obstacle problem ($\\gamma=1$), the Alt-Caffarelli/Bernoulli problem ($\\gamma=0$), and minimal surfaces as $\\gamma\\to-2$, so a common smoothness theorem anchors the whole family. The proof sends the solution through a hodograph transformation and reduces smoothness to Schauder estimates for a degenerate quasilinear PDE with a Neumann condition. In the negative-exponent regime the same regularity is used to compute the second variation and to show that stable axially symmetric cones are one-dimensional in low dimensions.","feed_headline":"Smooth free boundaries for every Alt-Phillips exponent","feed_subtitle":"A hodograph-Schauder proof covers negative exponents and rules out axially symmetric singular cones in low dimensions.","key_machinery":"The load-bearing object is the hodograph transform $h$: the inverse of $\\Phi(x',x_d)=(x',w(x',x_d))$, whose trace on $\\{x_d=0\\}$ is a local parametrization of the free boundary. The key identity is that the Alt-Phillips equations become the degenerate quasilinear system $\\operatorname{div}(x_d^s DF(\\nabla h))=0$ in the half-ball, with $F(p)=(|p|^2+1)/p_d$ and boundary condition $\\lim_{x_d\\to0^+} x_d^s DF(\\nabla h)\\cdot e_d=0$. Theorem 1.2, a Schauder estimate for uniformly convex $F$ in this weighted setting, upgrades $C^{1,\\alpha}$ to $C^{k,\\alpha}$ and then $C^\\infty$. On the stability side, the central identity is the second inner variation along the normal field $\\xi=(\\nabla w/|\\nabla w|)f$, which evaluates to $\\int_{\\Omega_w} w^s|\\nabla w|^2(|\\nabla f|^2-A_w^2 f^2)\\,dx$ with $A_w^2=|\\nabla^2 w|^2/|\\nabla w|^2 - |\\nabla^2 w\\nabla w|^2/|\\nabla w|^4$.","core_discovery":"The central claim, Theorem 1.1, is that if $u$ is a local minimizer of $J_\\gamma$ in $B_1$ and $x_0\\in\\partial\\Omega_u$ is a regular free-boundary point, then $\\partial\\Omega_u$ is locally the graph of a smooth function and both $w=\\beta u^{1/\\beta}$ and $u/\\operatorname{dist}(\\cdot,\\partial\\Omega_u)^\\beta$ are smooth in $\\Omega_u$ near $x_0$. The proof treats all $\\gamma\\in(-2,2)$ at once: with $s=\\beta\\gamma$, the hodograph transform $\\Phi(x',x_d)=(x',w(x',x_d))$ has an inverse $h$ whose trace parameterizes the free boundary, and $h$ solves the degenerate quasilinear equation $\\operatorname{div}(x_d^s DF(\\nabla h))=0$ with $F(p)=(|p|^2+1)/p_d$ and a Neumann boundary condition. Theorem 1.2 supplies Schauder estimates for such equations under uniform convexity of $F$, and iteration gives $C^\\infty$. For $\\gamma\\in(-2,0)$, Theorem 1.3 derives the stability inequality $\\int_{\\Omega_w} w^s|\\nabla w|^2(|\\nabla f|^2-A_w^2 f^2)\\,dx\\ge 0$, where $A_w^2$ encodes the second fundamental form of level sets, and Theorem 1.4 uses it to prove that axially symmetric minimizing cones are one-dimensional for $d\\le 6$, or for $d=7$ and $\\gamma<(10-8\\sqrt5)/11$.","pith_inferences":["Beyond the paper's claims, the hodograph-Schauder route looks portable to other degenerate one-phase free boundary problems whose boundary condition is encoded by a power weight; the authors do not pursue this.","Beyond the paper's claims, the stability inequality can be used as a spectral test on candidate cones, since $A_w^2$ is computable from the geometry of level sets; the paper does not run such tests.","Beyond the paper's claims, removing the uniform $C^{2,\\alpha}$ compactness assumption in the $\\gamma\\to-2$ limit would turn the convergence result into a statement about all stable solutions; that remains open here."],"forward_implications":["Regular free-boundary points are $C^\\infty$: all derivatives of the interface exist, so curvature and other higher-order geometric quantities are well defined there.","The stability inequality supplies one second-variation criterion for the whole negative-exponent regime and reduces to the known Alt-Caffarelli and minimal-surface criteria at the endpoints.","Axially symmetric stable or minimizing cones are one-dimensional in low dimensions ($d\\le 6$, and $d=7$ for $\\gamma$ below about $-0.7171$), so any singular minimizer there must be non-axial.","The eigenvalue criterion $\\lambda_s(\\Sigma_w)\\ge -((d+s-2)/2)^2$ gives an explicit spectral test for homogeneous cones and converges to the minimal-surface stability criterion as $\\gamma\\to-2$."],"supporting_citations":[{"why":"Establishes the baseline one-sided regularity and the free-boundary condition for negative exponents; Lemma 2.3 and the linearized iteration rest on it.","marker":"[20]"},{"why":"Supplies the Schauder theory for degenerate linear equations with integrable weights that Proposition 2.8 adapts to the quasilinear setting.","marker":"[47]"},{"why":"Introduces the hodograph transformation that converts free-boundary regularity into PDE boundary regularity.","marker":"[34]"},{"why":"Proves smoothness for positive exponents by a different method, the result that Theorem 1.1 extends and unifies.","marker":"[42]"},{"why":"Computes the second variation for positive exponents and provides the comparison for the negative-exponent stability condition.","marker":"[33]"},{"why":"Gives the Alt-Caffarelli stability inequality and low-dimensional rigidity that Theorem 1.3 generalizes.","marker":"[7]"},{"why":"Provides the stable-minimal-cone eigenvalue criterion recovered in the gamma-to-minus-two limit.","marker":"[44]"},{"why":"Proves compactness and Gamma-convergence to perimeter as the exponent tends to minus two, used in the asymptotic stability analysis.","marker":"[19]"}],"fun_headline_variants":["Alt-Phillips free boundaries are always smooth","Smooth free boundaries, no singular cones for Alt-Phillips","All Alt-Phillips free boundaries smooth, stable cones ruled out","One proof covers every Alt-Phillips exponent, kills singular cones","Hodograph proof: smooth boundaries, no axially symmetric cones"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 2.3: near a regular free-boundary point every local minimizer is a regular solution in the sense of Definition 2.1, meaning it has Hölder regularity with $\\alpha>-s$ and satisfies the weighted boundary condition pointwise; the proof of this lemma is a compressed linearization-and-iteration argument that cites [20, Prop. 7.2] instead of carrying out the full bootstrap.","fun_headline_variants_meta":{"raw":{"variants":["Alt-Phillips free boundaries are always smooth","Smooth free boundaries, no singular cones for Alt-Phillips","All Alt-Phillips free boundaries smooth, stable cones ruled out","One proof covers every Alt-Phillips exponent, kills singular cones","Hodograph proof: smooth boundaries, no axially symmetric cones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000557,"raw_usage":{"total_tokens":2702,"prompt_tokens":1052,"completion_tokens":1650,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":668,"completion_tokens_details":{"reasoning_tokens":1563}},"tokens_in":668,"tokens_out":1650,"duration_ms":14011,"temperature":1.0,"reasoning_tokens":1563,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:34:02.675289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check would be to construct a solution of $\\operatorname{div}(x_d^s DF(\\nabla v))=0$ with $F$ uniformly convex and $F\\in C^{3,\\alpha}$ that is $C^{1,\\alpha}$ but not $C^{2,\\alpha}$ in $B^+_{1/2}$; such a counterexample would falsify Theorem 1.2 and with it the smoothness claim. On the stability side, evaluating inequality (4.2) on any explicit nontrivial axially symmetric 1-homogeneous cone in dimension 6 would settle Theorem 1.4.","supporting_citations":[{"cited_title":"De Silva and O","cited_arxiv_id":null,"evidence_quote":"Establishes the baseline one-sided regularity and the free-boundary condition for negative exponents; Lemma 2.3 and the linearized iteration rest on it."},{"cited_title":"Terracini, G","cited_arxiv_id":null,"evidence_quote":"Supplies the Schauder theory for degenerate linear equations with integrable weights that Proposition 2.8 adapts to the quasilinear setting."},{"cited_title":"Kinderlehrer and L","cited_arxiv_id":null,"evidence_quote":"Introduces the hodograph transformation that converts free-boundary regularity into PDE boundary regularity."},{"cited_title":"Restrepo and X","cited_arxiv_id":null,"evidence_quote":"Proves smoothness for positive exponents by a different method, the result that Theorem 1.1 extends and unifies."},{"cited_title":"Global energy minimizers for free boundary problems and full regularity in three dimensions","cited_arxiv_id":null,"evidence_quote":"Gives the Alt-Caffarelli stability inequality and low-dimensional rigidity that Theorem 1.3 generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the stable-minimal-cone eigenvalue criterion recovered in the gamma-to-minus-two limit."},{"cited_title":"De Silva and O","cited_arxiv_id":null,"evidence_quote":"Proves compactness and Gamma-convergence to perimeter as the exponent tends to minus two, used in the asymptotic stability analysis."}],"review_version":1}