{"id":"5afb5855-af92-4a8b-b3df-d7cd48c700b0","arxiv_id":"2509.02064","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Alt-Phillips problem with γ in (1,2), the free boundary touches the fixed boundary tangentially wherever the Dirichlet data vanish, in the fully nonlinear and in the linear case.","lead":"This paper proves that for the fully nonlinear Alt-Phillips free boundary problem with parameter γ between 1 and 2, the free boundary always meets the fixed boundary tangentially where the boundary data vanish. The result is new even when the operator is the classical Laplacian, and it completes the boundary-touch picture for the parameter range currently understood.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.1 is false as stated: the 1D family u_c(x)=g_c(x_n), (g_c')^2=c^2+2g_c^γ, g_c(0)=0, has Γ(u_c)=∅, satisfies (i)-(ii), but is not the claimed profile; the proof of Theorem 1.1 needs an additional hypothesis.","rationale":"The reader identified the missing regularity input u=|∇u|=0 on Γ as the weakest assumption, and that is related, but the more concrete and decisive problem is that Lemma 4.1 is false under its stated hypotheses. The one-parameter family u_c(x)=g_c(x_n) with nonzero normal derivative at the fixed boundary satisfies all displayed hypotheses: it solves the Laplacian Alt-Phillips equation in the half-space, vanishes on the fixed boundary, has no interior free boundary so the overdetermined hypothesis is vacuous, and satisfies the growth nondegeneracy from below. It is not the claimed one-dimensional profile. This is an explicit counterexample, not a missing citation. The proof of Lemma 4.1 fails exactly at (4.22): the uniform lower bound on ∂e u_Rj over {u_Rj>η} cannot follow from C^1 convergence because the level set may approach the fixed boundary for a function with nonzero normal derivative there. The theorem may still be true, and the actual blow-up limits probably have the extra property |∇u0(0)|=0, but that property is not in Lemma 4.1 and the current proof does not use it. Therefore the manuscript as written has a serious gap in its central classification step; it should be conditionally accepted only after Lemma 4.1 is corrected and the proof of (4.22) is repaired with a boundary lower bound argument.","tokens_in":14921,"tokens_out":37209,"duration_ms":402788,"concrete_test":"Take γ=3/2 (so β=4) and numerically solve g''=(3/2)g^{1/2}, g(0)=0, g'(0)=1, then set u(x)=g(x_n). Verify that u satisfies (4.1), that (i) holds with a=min{1,(√2/4)^4}/2, that (ii) is vacuously true since Γ(u)=∅, and that u(1,1)≠(√2/4)^4. This directly falsifies Lemma 4.1 as written. To test the repair, add the hypothesis |∇u|=0 on R^n_0 and check whether the proof of (4.22) can be made to work; if the level set {u_Rj>η} can still approach x_n=0, the repair is insufficient.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Under the stated definition Γ(u)=∂{u>0}∩{xn>0}, Lemma 4.1 cannot exclude the explicit half-space solutions u_c(x)=g_c(x_n), c>0, where g_c solves g_c''=γg_c^{γ-1}, g_c(0)=0, g_c'(0)=c. The first integral gives (g_c')^2/2 - g_c^γ = c^2/2, so g_c is smooth on (0,∞) and C^{1,α} up to 0, positive for x_n>0, zero on x_n=0. Thus u_c∈C^{1,α}_{loc}(R^n_+), ∆u_c=γu_c^{γ-1}, u_c=0 on R^n_0, and u_c≥0. Moreover Γ(u_c)=∅, so the overdetermined condition in (ii) is vacuous, and (i) holds with a=min{c,(√2/β)^β}/2 because g_c(t)≥ct for small t and g_c(t)∼(√2 t/β)^β for large t. But u_c differs from (√2(x_n/β)_+)^β. Hence Lemma 4.1 as stated is false. This is not a mere notation slip: the proof of (4.22) asserts a uniform δ>0 for ∂e u_Rj on {u_Rj>η}; for u_c, ∂e u_Rj=R_j^{1-β}g_c'(R_j x_n) tends to 0 on the level set {u_Rj>η}, which approaches x_n=0 as R_j→∞. The argument requires an extra hypothesis such as |∇u(0)|=0 on the fixed boundary (true for the actual blow-up limits via Lemma 3.1) plus a boundary Harnack/monotonicity input ensuring {u_Rj>η} stays away from x_n=0; neither is stated or proved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the fully nonlinear Alt–Phillips problem in the upper half-ball with zero Dirichlet data on the fixed boundary, for 1 < γ < 2. It claims that the free boundary Γ(u) = ∂{u>0} ∩ {x_n>0} meets the fixed boundary tangentially wherever it touches points where the Dirichlet data vanish. The proof proceeds by rescaling at a contact point and proving a classification lemma (Lemma 4.1) for global solutions of the limiting classical problem Δu = γu^{γ-1} in the half-space: the asserted conclusion is that every such solution with the stated growth and nondegeneracy assumptions is the one-dimensional profile (√2 (x_n/β)_+)^β, β = 2/(2-γ). From this, the authors conclude that blow-up limits have empty free boundary, giving the tangential-touch statement. The argument uses a Weiss-type monotonicity formula, an improvement-of-monotonicity lemma, and boundary regularity estimates.","tokens_in":15224,"tokens_out":17517,"duration_ms":194024,"significance":"If the main theorem and the supporting classification were correct, the result would be a meaningful advance: the tangential-touch property is new for γ ∈ (1,2) even for the Laplacian, and the strategy of reducing boundary contact to a classification of global half-space solutions is attractive. The paper also offers a fully nonlinear extension of a line of results known previously only for γ = 0 and γ = 1. However, the central classification lemma is false as stated, and there are substantial unproved regularity assertions about the free boundary. The core idea is promising and likely repairable, but the manuscript in its current form cannot be accepted.","major_comments":[{"comment":"Lemma 4.1 is false as stated. For any c > 0, let g_c solve g_c'' = γ g_c^{γ-1}, g_c(0)=0, g_c'(0)=c. The first integral gives (g_c')^2/2 - g_c^γ = c^2/2, so g_c is positive and smooth on (0,∞), C^{1,α} up to 0, and u_c(x) = g_c(x_n) solves (4.1). For this u_c, Γ(u_c) = ∅, so hypothesis (ii) is vacuous, and (i) holds with a = min(c/2, (√2/β)^β/2) because g_c(t) ≥ ct for small t and g_c(t) ∼ (√2 t/β)^β for large t. Yet u_c is not the claimed profile. The proof's final step, 'Then (4.16) follows immediately', is invalid because the ODE g''=γg^{γ-1} with g(0)=0 has a one-parameter family of solutions g_c. The classification needs an additional hypothesis, e.g. |∇u(0)|=0, which is available for the blow-up limits used in the proof of Theorem 1.1 through Lemma 3.1. This false lemma is load-bearing for the advertised uniqueness claim and for the identification of the blow-up limit in the proof","section":"Section 4, Lemma 4.1 and the paragraph after (4.21)"},{"comment":"The assertion 'This implies an overdetermined condition on the free boundary, namely, u = |∇u| = 0 on Γ(u)' is not a consequence of Evans–Krylov regularity alone. Evans–Krylov gives classical solvability in {u>0}, not regularity up to the free boundary Γ(u). This overdetermined condition is used throughout: in Lemma 3.1, in condition (3.1) of Lemma 3.2, and in the compactness argument leading to (3.16). If this is a theorem proved in the authors' earlier work [WY], it must be cited explicitly at the point of use with the exact statement; otherwise it must be proved in this paper. As it stands, the regularity input is missing and is a load-bearing gap.","section":"Section 2.2, the paragraph beginning 'Let u be a continuous viscosity solution'"},{"comment":"In the proof of the growth estimate, the authors write 'Due to the above and (3.10), D u_j(x_j)=0, D^2 u_j(x_j)=0'. The bound (3.10) is a C^{2,α} estimate in B^+_{3/4}, which does not by itself imply that the full Hessian vanishes at an interior free-boundary point x_j. The condition u_j=0 on one side of Γ(u_j) forces the tangential second derivatives to vanish, but not necessarily the normal second derivative. Therefore the Taylor estimate sup_{B_{2^{-k}}} u_j ≤ C2^{-k(2+α)} in (3.11) is not justified. This step is used to rescale the equation to obtain the limiting harmonic function in (3.12), so the gap affects the entire compactness argument. The missing input is an additional free-boundary regularity statement (vanishing of D^2 at free-boundary points) that is neither proved nor cited.","section":"Section 3, proof of Lemma 3.2, around (3.10)–(3.11)"}],"minor_comments":[{"comment":"The notation 'eC0' is confusing. It appears to denote a large constant, but the exponential notation invites misreading. Please introduce a named constant, e.g. K, and state the choices of K and c0 explicitly.","section":"Section 2.3, Lemma 2.7"},{"comment":"There are several typos in this proof: 'yn ≥ −ln' should be 'yn ≥ −l_j' in multiple places, and 'Arzalà-Ascoli' in Section 3 should be 'Arzelà–Ascoli'. These do not affect the mathematics but should be corrected.","section":"Section 3, proof of Lemma 3.2"},{"comment":"In Case 2 of Step 1, the text reads 'there exists r1(p) such that ∂xn u(p)=0 in ...'; the argument of the function should be x, not p. Please fix.","section":"Section 4, Lemma 4.3"},{"comment":"In the contradiction argument, the same symbol u is used for the fixed solution and for the sequence u_j. After passing to a subsequence, the rescaling of u_j at the origin should be explicitly distinguished; the invocation of (3.16) requires uniform compactness for the sequence, which should be stated.","section":"Proof of Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper has a promising strategy and the main theorem may well be correct, but the current manuscript contains a false classification lemma and unproved free-boundary regularity assertions. The false examples u_c are excluded by the boundary gradient condition |∇u(0)|=0 that is already available for the actual blow-up limits, so I believe the issues are fixable within the scope of a revision rather than fatal. The authors should restate Lemma 4.1 with the correct hypothesis, repair the proof of the ODE reduction, and either prove or precisely cite the free-boundary C^1/C^2 regularity used in Section 3."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the main theorem is new and probably true, but Lemma 4.1 is false as stated. The one-dimensional ODE has a family u_c(x)=g_c(x_n), c>0, with g_c''=γg_c^{γ-1}, g_c(0)=0, g_c'(0)=c. For these, Γ(u_c)=∅, so condition (ii) is vacuous, and condition (i) holds with a constant depending on c. None equals (√2(x_n/β)_+)^β. The stress-test's counterexample is right; its explanation about (4.22) failing on the level set is not—for fixed η the level set for u_c stays away from x_n=0 and ∂_{x_n}u_c,R has a positive uniform limit. The real issue is the last step of Lemma 4.1 ('repeating the proof as in Lemma 4.4'): monotonicity in all upper directions gives u=u(x_n), but the ODE still has the c-family. The missing input is |∇u(0)|=0; in the actual application the blow-up limit has exactly this (Lemma 3.1 plus C^{1,α} convergence), so the proof of Theorem 1.1 can be repaired by adding that hypothesis to Lemma 4.1.\n\nThe paper's positives: the result is genuinely new for γ∈(1,2), including the Laplacian; the scaling β=2/(2-γ) and the barrier in Lemma 2.7 check out; no fitted parameters; the Weiss formula and blow-up scheme are appropriate. The survey of known boundary-touch results is accurate. The reader's concern about the overdetermined condition is also valid: Section 2.2 asserts u=|∇u|=0 on Γ(u) as if it followed from Evans-Krylov, but it needs the [WY] regularity theory, and it should be cited at the point of use. Lemma 3.2's D^2u(x_j)=0 also deserves a line. Notation slips are minor.\n\nBottom line: serious paper, deserves a referee, but the current version has a false lemma and a proof gap that are fixable. I'd send it back for revision.","headline":"Genuinely new tangency result for γ∈(1,2), but Lemma 4.1 is false as stated; the main theorem is likely salvageable with an extra hypothesis that the application already provides.","tokens_in":15979,"tokens_out":16694,"would_cite":true,"duration_ms":172561,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R35","35J60","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the fully nonlinear Alt-Phillips problem with γ in (1,2), every contact between the free boundary and the fixed boundary is tangential, and the proof is a uniqueness classification of global half-space solutions.","keywords":["Alt-Phillips problem","free boundary","fixed boundary","tangential contact","fully nonlinear elliptic equations","blow-up analysis","global classification","degenerate semilinear equation"],"falsifier":"Compute, for n=2, γ=3/2 and F=Δ, the exact or numerical solution of (1.1) with 0 on the free boundary and inspect the rescaled sequence r^{-β}u(rx). If any subsequential limit is not ((√2/β)(x_n)_+)^β, or if rescaled free-boundary points accumulate along a positive angle to {x_n=0}, Theorem 1.1 fails. More directly, any nonzero global half-space solution of Δu=γu^{γ-1} with r^β growth and nondegeneracy different from the explicit profile falsifies Lemma 4.1.","tokens_in":14590,"feed_emoji":"📐","tokens_out":11493,"duration_ms":107201,"temperature":0.7,"pith_summary":"This paper proves that, for the fully nonlinear Alt-Phillips problem with parameter γ in (1,2), the free boundary—the boundary of the region where the solution is positive—meets the fixed boundary only tangentially at points where the Dirichlet data vanish. The result is new even for the Laplacian. The proof rescales a solution at a contact point and shows that every limit of the rescaled solutions is the same explicit one-dimensional profile, namely u(x)=((√2/β)(x_n)_+)^β with β=2/(2−γ). Because that profile's positive region stays strictly away from the fixed boundary in the limit, the free boundary approaching the contact must flatten onto the fixed boundary. The classification of the limiting profile is the core of the argument.","feed_headline":"Alt-Phillips free boundary and wall meet at zero angle","feed_subtitle":"For every γ in (1,2), including the Laplacian, the contact is forced by a unique one-dimensional limit profile.","key_machinery":"The engine is the blow-up analysis tied to a uniqueness classification. With β=2/(2−γ), rescale u near a contact point by u_r(x)=u(rx)/r^β. Growth estimates give a uniform upper bound, nondegeneracy gives a lower bound, so rescaled solutions converge along subsequences; the fully nonlinear operator degenerates to the Laplacian in the limit, and the combination of monotonicity and improvement-of-monotonicity forces the limit to be β-homogeneous. The classification then shows that the only β-homogeneous half-space solution of Δu=γu^{γ-1} with zero boundary data is the explicit one-dimensional profile. This one profile has no free-boundary contact in the limit, and that absence of contact is wh","core_discovery":"The central claim is Theorem 1.1: if F is a uniformly elliptic, convex, C^1 fully nonlinear operator with F(0)=0 and trace normalization, γ∈(1,2), and u solves the Alt-Phillips system (1.1) with 0 a free-boundary point on the fixed boundary, then the free boundary near 0 is contained in {x_n ≤ σ(|x|)|x|} for a universal modulus σ tending to 0. In other words, contact is tangential. The proof rests on Lemma 4.1, which classifies all global solutions of the limiting classical problem Δu=γu^{γ-1} in the half-space with zero boundary data and the natural growth and nondegeneracy bounds: the only such solution is the one-dimensional explicit profile. Blowing up any solution at a contact point yie","pith_inferences":["The same mechanism likely governs energy minimizers of the Alt-Phillips functional in this range, since the argument uses the PDE rather than minimality; this would unify interior and boundary regularity for minimizers.","Because the limiting profile is explicit, a full asymptotic expansion of u near the contact point could be obtained by linearizing around that profile; the overdetermined condition u=|∇u|=0 at the free boundary would fix the expansion coefficients.","The paper leaves γ∈(0,1) open, and the stated reasons—no convexity of global solutions and a continuum of homogeneous solutions—suggest that a direct extension of this classification will require new stability inputs rather than a rerun of the same proof.","A straightforward numerical check for the Laplacian at γ=3/2 in two dimensions could test the claimed universality: the rescaled free boundary should approach the fixed boundary with the slope bound σ(|x|)|x|, and all blow-ups should be the explicit profile."],"forward_implications":["Every free-boundary point on the fixed boundary is a tangential contact, for the Laplacian and for every operator in the class; no free boundary can enter the domain at a positive angle.","At a contact point, every blow-up of the solution is the same explicit power profile, so the leading-order shape of the solution near the contact is universal.","The flatness of the free boundary is controlled by a universal modulus σ(|x|) that depends only on dimension, γ, ellipticity, and the operator's modulus of continuity, not on the individual solution.","This completes the picture for the Laplacian in the range γ∈(1,2), where no boundary-touch result existed before, and matches the classical tangential-contact results for γ=0 and γ=1.","The classification of global half-space solutions implies rigidity: any half-space solution with the right growth and nondegeneracy is one-dimensional, so the free boundary cannot branch from the fixed boundary."],"supporting_citations":[{"why":"Supplies the viscosity-solution framework, interior regularity, and the stability lemma used to pass fully nonlinear operators to the Laplacian in blow-up limits.","marker":"[CC]"},{"why":"Provides the regularity theory for the fully nonlinear Alt-Phillips problem for this exponent range, cited as the source of the overdetermined free-boundary condition and convexity of global solutions.","marker":"[WY]"},{"why":"Provides the change of variables w=u^{2/β} and the transformed equation used in the monotonicity step of the classification.","marker":"[AP]"},{"why":"Supplies the boundary monotonicity formula that forces blow-down limits to be β-homogeneous.","marker":"[W1]"},{"why":"Supplies the lemma that blow-up limits of homogeneous functions are invariant in the direction of the blow-up point, used to control derivatives near the fixed boundary.","marker":"[V]"},{"why":"Provides boundary C^{1,α} and C^{2,α} regularity estimates for fully nonlinear equations, used for compactness of rescaled solutions.","marker":"[LZ]"},{"why":"Provides boundary regularity for viscosity solutions, used in the growth estimate and the compactness argument.","marker":"[SS]"},{"why":"Observes the w-transformation lemma together with [AP], giving the equation for v_n in the classification proof.","marker":"[B2]"}],"fun_headline_variants":["Alt-Phillips: free boundary touches wall tangentially for gamma in (1,2)","Fully nonlinear Alt-Phillips: tangential free-boundary contact for gamma in (1,2)","For gamma in (1,2), Alt-Phillips free boundary meets wall tangentially","Alt-Phillips: tangential contact with wall proven for gamma in (1,2)","Tangential touch at fixed boundary for fully nonlinear Alt-Phillips"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that every solution is differentiable up to the free boundary with both the value and the gradient vanishing there; if that boundary regularity were false for some γ∈(1,2) or some operator F, the classification of blow-up limits would not apply to actual solutions.","fun_headline_variants_meta":{"raw":{"variants":["Alt-Phillips: free boundary touches wall tangentially for gamma in (1,2)","Fully nonlinear Alt-Phillips: tangential free-boundary contact for gamma in (1,2)","For gamma in (1,2), Alt-Phillips free boundary meets wall tangentially","Alt-Phillips: tangential contact with wall proven for gamma in (1,2)","Tangential touch at fixed boundary for fully nonlinear Alt-Phillips"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00116,"raw_usage":{"total_tokens":4573,"prompt_tokens":606,"completion_tokens":3967,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":350,"completion_tokens_details":{"reasoning_tokens":3855}},"tokens_in":350,"tokens_out":3967,"duration_ms":30287,"temperature":1.0,"reasoning_tokens":3855,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:59:35.441265+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for n=2, γ=3/2 and F=Δ, the exact or numerical solution of (1.1) with 0 on the free boundary and inspect the rescaled sequence r^{-β}u(rx). If any subsequential limit is not ((√2/β)(x_n)_+)^β, or if rescaled free-boundary points accumulate along a positive angle to {x_n=0}, Theorem 1.1 fails. More directly, any nonzero global half-space solution of Δu=γu^{γ-1} with r^β growth and nondegeneracy different from the explicit profile falsifies Lemma 4.1.","supporting_citations":[],"review_version":1}