{"id":"402ebd95-ef12-43fe-b9ec-1c77ac457ee8","arxiv_id":"2411.18710","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims an existence theorem for a general free boundary problem on stratified Lie groups, but the variational proof contains critical gaps.","lead":"This paper claims to prove that a class of free boundary equations has a solution on curved spaces called stratified Lie groups. The proof has serious gaps, so the result is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Mountain-pass construction is run in the wrong λ-regime: the only negative-energy point is obtained from E(u)→−∞ as λ→∞, while Theorem 1.1 claims 0<λ<λ*; for small λ the path set Γ_ε may be empty.","rationale":"The reader's weakest_assumption identifies the unproved monotonicity lemma and missing L∞ bound. Those are genuine gaps, but they are less central than the λ-regime problem: Lemma 3.1 is quoted from a published paper and might be accepted as a black box, and the L∞ bound could conceivably be repaired by Moser iteration. The mountain-pass geometry, however, is the actual engine of the existence proof, and in the paper it is explicitly run in the wrong direction: the only justification for a negative-energy point is the limit E(u)→−∞ as λ→∞. This cannot yield the claimed small-λ theorem. The radial Euclidean computation reinforces the suspicion by showing that even the symmetric candidate problem has no solution for arbitrarily small λ. Therefore the stated range in Theorem 1.1 is not supported; the proof would need either a quantitatively different mountain-pass geometry for small λ or a corrected theorem with λ large. I keep the reader's REJECT verdict but with a different primary emphasis.","tokens_in":11212,"tokens_out":15172,"duration_ms":156163,"concrete_test":"Take G=R^N (the abelian stratified Lie group), Ω=B_R, g≡1, and solve the radial version of (1): on {u>1}=B_a set u=1+λ(a²−r²)/(2N), on a √2, so no positive solution exists for λ < N√2/R. Thus the small-λ assertion is both unproved and implausible; the proof would at best support a large-λ existence statement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a variational framework for a Bernoulli-type free boundary problem on stratified Lie groups. The nonsmooth energy E(u)=∫(|∇_G u|²/2 + χ_{u>1} − λ G(x,(u−1)_+)) is approximated by C¹ functionals E_ε, and the author claims that mountain-pass critical points of E_ε converge to a positive solution satisfying the equation in Ω\\F(u), the free-boundary condition in the viscosity sense, and u=0 on ∂Ω. The main theorem states that such a solution exists for all 0<λ<λ*, assuming the subcritical growth condition (g1) and positivity (g2). The proof relies on a quoted monotonicity lemma of Choudhuri–Repovš to obtain uniform Lipschitz estimates and on a Palais–Smale condition for E_ε.","tokens_in":11434,"tokens_out":6831,"duration_ms":120037,"significance":"If the main theorem were correct, it would provide a meaningful extension of Perera's Euclidean result [24] and of the power-nonlinearity result of Choudhuri–Repovš [10] to general subcritical nonlinearities on stratified Lie groups. The variational approximation idea is natural, and the paper correctly identifies the monotonicity lemma as the key external tool. However, the central existence proof has serious gaps, and the claimed small-λ range is contradicted by an explicit admissible example, so the contribution as it stands is not established.","major_comments":[{"comment":"The mountain-pass geometry is established in the wrong λ-regime. The proof notes that E(u)→−∞ as λ→∞ and then asserts the existence of u0 with E_ε(u0)<0; this only supplies a negative endpoint for large λ, whereas Theorem 1.1 claims existence for 0<λ<λ*. For small λ, E_ε is bounded below and, under (g1), can be nonnegative for all u once λ is below a threshold depending on the Sobolev constants, so the path class Γ_ε may be empty. This is not merely a missing argument: in the Euclidean radial case g≡1, Ω=B_R, the free-boundary condition forces (λa/N)² − |∇u_-|² = 2 with a N√2/R, so no solution exists for small λ. Thus the stated small-λ theorem is contradicted by an admissible example.","section":"§4, proof of Theorem 1.1"},{"comment":"The convergence lemma assumes only boundedness of (u_j) in W^{1,2}_0(Ω)∩L∞(Ω), but the proof later asserts that the sequence is bounded in C^{2,α}-norm and uses C² convergence near ∂Ω. No such estimate is proved or follows from the assumed bounds, since the right-hand side of (7) contains ε_j^{-1}β((u_j−1)/ε_j). The uniform Lipschitz estimate obtained from Lemma 3.1 is also not justified: the hypotheses of Lemma 3.1, including the structural condition F(t)=o(t²), are not verified for the sequence u_j uniformly in j, and the term (2/ε_j)χ_{|u_j−1|<ε_j} cannot be handled by the lemma as stated. Without these estimates, the uniform convergence, the local C¹ convergence, and the free-boundary convergence in (i)–(iv) are unsupported.","section":"§3, Lemma 3.2"},{"comment":"The proof of the Palais–Smale condition is invalid as written. The displayed inequality ≤ ||u_n||² − λ∫g(x,(u_n−1)_+)dx + (2/ε)|Ω| omits the nonnegative term (1/ε)∫β((u_n−1)/ε)u_n dx coming from the derivative of the penalization, so it cannot be used to control ||u_n||. The subsequent transition from E'_ε(u_n)→0 to strong convergence uses the very boundedness that was to be proved; the sentence 'A standard argument implies that (u_n) is bounded' is not substantiated, and the positive zeroth-order term in the approximating equation works against coercivity. Therefore Lemma 4.1 is not established.","section":"§4, Lemma 4.1"},{"comment":"The passage from equations (14)–(18) to the conclusion that u satisfies the free-boundary condition in the viscosity sense is not rigorous. The integrals in (18) are over level sets {u=1±ε±}, while the claimed limit involves an identity on F(u), and the formula for the normal vector n=±∇u/|∇u| is asserted without regularity of the level set {u=1}. Moreover, the signs in (18) are inconsistent: the positive-level integral contains (1−1/2|∇u|²) while the negative-level integral contains −(1/2|∇u|²), so the displayed cancellation is unclear. Since Theorem 1.1 explicitly includes the free-boundary condition, this step is load-bearing.","section":"§3, free-boundary derivation"}],"minor_comments":[{"comment":"In Definition 2.2, N1 is used for the dimension of the first stratum, but later the text says 'we let N1=N' for the number of generators; this conflicting notation should be clarified.","section":"§2"},{"comment":"The keyword 'startified Lie group' is a typo for 'stratified Lie group'; the abstract also contains several typographical errors such as 'exits' for 'exists' and 'founded' for 'found'.","section":"Abstract and keywords"},{"comment":"The proof of Lemma 3.2 is headed 'Proof of Lemma 3.1'; the lemma labels should be corrected.","section":"§3"},{"comment":"In the proof of Lemma 4.1, the growth condition (g1) is stated with exponent 1 with an index i that is not defined, and the notation dn for the surface measure should be made explicit.","section":"§3, equation (11)"}],"recommendation":"reject","confidential_remarks":"The small-λ existence claim in Theorem 1.1 appears to be false already for g≡1 on a Euclidean ball, and the variational proof in §4 establishes at best a large-λ mountain-pass geometry. Even if the theorem were reformulated for large λ, Lemma 3.2 and Lemma 4.1 would require substantial new estimates that are not present in the manuscript. The paper is not suitable for publication in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the existence theorem as stated is not proved and looks false. The proof in §4 creates a negative-energy point by noting E(u)→−∞ as λ→∞, which gives a mountain-pass path for large λ. But Theorem 1.1 claims 0<λ<λ*. For small λ, the approximating energy is nonnegative on all of W^{1,2}_0, so the path class Γ_ε is empty. The stress-test's radial example is the right way to see this: for g≡1, Ω=B_R, a radial solution with {u>1} a ball of radius a must satisfy λ²a²/N² − (u_−'(a))² = 2, hence λa > N√2, so no solution for λ < N√2/R. So the small-λ statement cannot be right.\n\nWhat the paper does well: the idea of approximating the non-smooth χ_{u>1} by a smooth B((u−1)/ε) and passing to the limit via a monotonicity lemma is the natural approach, and the setting (general subcritical g on stratified Lie groups) is a legitimate extension of Perera [24] and Choudhuri–Repovš [10]. The convergence lemma attempts the right program: uniform convergence, local C^1, strong W^{1,2}, and an energy inequality.\n\nThe soft spots beyond the λ-regime are real. The Palais–Smale proof in Lemma 4.1 is not valid as written: it drops the positive (1/ε)β term and asserts a 'standard argument' without controlling it. Lemma 3.2 assumes the sequence is bounded in L∞ and that the approximating critical points are bounded in C^{2,α}; neither bound is established before the lemma is applied. The monotonicity lemma of [10] is quoted without proof, and the application writes the RHS as (2/ε)χ_{|u−1|<ε} without the F(|∇u|) factor, so a reader must take the lemma's exact hypotheses on faith. There are also many typos and passages that read as drafts.\n\nNet: the paper has a plausible skeleton but the main theorem contradicts its own proof, and in simple cases the theorem is false. A revised version that claims existence for large λ might be worth a look, but this manuscript would need substantial reworking of the PS proof and the convergence argument, plus a corrected statement.\n\nRecommendation: I'd still send it to a referee rather than desk-reject — the connection to Perera and to stratified Lie groups is real, and a referee can quickly confirm the λ-regime problem — but I'd expect the recommendation to be reject and resubmit, not accept.","headline":"The theorem claims existence for small λ, but the proof's mountain-pass endpoint only works for large λ, and a radial example shows the small-λ statement is false.","tokens_in":11941,"tokens_out":8823,"would_cite":false,"duration_ms":78277,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34R35","35J25","35J20","35B38"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, under mild growth assumptions on the nonlinearity, a class of elliptic free boundary problems on stratified Lie groups admits a positive solution for small λ, with the free boundary condition holding in the…","keywords":["Sub-Laplacian","stratified Lie group","free boundary problem","mountain pass","viscosity solution","Heisenberg group","subcritical nonlinearity","horizontal gradient"],"falsifier":"Construct, on the unit ball of a stratified Lie group, a sequence u_j of Lipschitz functions satisfying the distributional inequalities of Lemma 3.1 with ε_j → 0 but with |∇_G u_j| unbounded; such an example would invalidate the monotonicity estimate that the proof uses. Alternatively, exhibit a bounded domain and a nonlinearity g obeying (g1)–(g2) for which the mountain pass critical points of E_ε are not uniformly bounded in L^∞ as ε → 0.","tokens_in":10978,"feed_emoji":"📐","tokens_out":5582,"duration_ms":47819,"temperature":0.7,"pith_summary":"The paper proves that a class of elliptic free boundary problems—where the unknown solves a sub-Laplacian equation away from the set where it crosses the level u = 1, with a jump condition on the horizontal gradient across that set—has a positive solution on any bounded domain in a stratified Lie group. The nonlinearity g is only assumed to grow sublinearly, g(x, s) ≤ α + β $s^{{m−1}}$ with 1 < m < 2, and to be positive. The proof constructs solutions as limits of mountain pass critical points of smooth approximating functionals, with the free boundary condition obtained in the viscosity sense. If correct, this extends existence results previously known for the Euclidean Laplacian and for the special power nonlinearity to general subcritical nonlinearities in non-Euclidean settings such as the Heisenberg group.","feed_headline":"Free boundary solutions exist on stratified Lie groups","feed_subtitle":"A variational approximation scheme now handles general subcritical nonlinearities beyond Euclidean space.","key_machinery":"The load-bearing structure is the approximating family of $C^{1}$ functionals E_ε(u) = ∫_Ω [½|∇_G u|^2 + B((u−1)/ε) − λ G_ε(x, (u−1)_+)] dx, where B is a smooth step function replacing the characteristic function χ_{u>1} and G_ε is the truncated primitive of g. Mountain pass critical points u_ε of E_ε solve the regularized equation (6), and convergence to a solution of (1) is driven by the monotonicity lemma (Lemma 3.1, quoted from the literature) which gives uniform horizontal gradient bounds for functions satisfying the distributional inequalities ±Lu ≤ (λ/ε)χ_{|u−1|<ε} F(|∇_G u|) + A. These bounds make the family equi-Lipschitz on compact sets, so Ascoli–Arzelà yields a uniform limit u, and a localized integration-by-parts argument forces the viscosity free boundary condition.","core_discovery":"The central claim is Theorem 1.1: under assumptions (g1) and (g2), there is a λ* > 0 such that for every 0 < λ < λ*, the free boundary problem (1) admits a positive solution u ∈ $W^{{1,2}}$_0(Ω) ∩ $C^{2}$(Ω \\ F(u)) that satisfies −Lu = λ χ_{u>1} g(x, (u−1)_+) classically away from the free boundary F(u) = ∂{u > 1}, the free boundary condition |∇_G u_+|^2 − |∇_G u_−|^2 = 2 in the viscosity sense, and u = 0 on ∂Ω. The solution is obtained as the limit of critical points u_j of smooth $C^{1}$ approximations E_{ε_j} of the non-smooth energy E, and the convergence is strong enough to pass to the limit in both the PDE and the free boundary condition.","pith_inferences":["The same variational approximation should yield multiple solutions by higher critical point theory, mirroring the Euclidean result, since the mountain pass level c_ε is bounded above by a level associated with the unregularized energy.","One could test numerically whether the viscosity free boundary condition is actually strong, for example by solving the regularized problems on the Heisenberg group and measuring convergence of the level sets {u_ε = 1} to a C^2 surface.","If the monotonicity lemma can be sharpened to give a rate, the method might extend to degenerate or nonlinear sub-Laplacians, such as p-sub-Laplacian analogues, where uniform gradient bounds are harder to obtain."],"forward_implications":["For every subcritical nonlinearity growing like s^{m−1} with 1 < m < 2, the free boundary problem (1) has a positive solution for all sufficiently small λ, not just for the power case previously treated.","The solution is Lipschitz on compact subsets and classical away from the free boundary, and the free boundary condition holds in the viscosity sense, matching the regularity obtained in Euclidean settings.","The convergence of approximating critical points is strong in W^{1,2}_0, so the energy of the limit is controlled up to the measure of the level set {u = 1}.","The result applies to every stratified Lie group, in particular the Heisenberg group, so the existence theory is non-Euclidean in a genuinely sub-Riemannian sense."],"supporting_citations":[{"why":"Supplies the monotonicity lemma in the stratified Lie group setting and treats the power-nonlinearity case that this paper generalizes.","marker":"[10]"},{"why":"Provides the Euclidean existence result for the same class of free boundary problems, which the theorem extends.","marker":"[24]"},{"why":"Contains the classical Euclidean monotonicity theorem underlying the non-Euclidean version used here.","marker":"[7]"},{"why":"Establishes the variational two-phase free boundary framework that motivates the energy and problem formulation.","marker":"[2]"},{"why":"Gives the divergence theorem for sub-Laplacians on stratified Lie groups used in the integration-by-parts arguments.","marker":"[25]"},{"why":"Provides the classical elliptic regularity theory used to obtain W^{2,2}_{loc} regularity and C^1 convergence away from the free boundary.","marker":"[21]"}],"fun_headline_variants":["Free boundary solutions now proven on Lie groups","Non-Euclidean free boundary problem solved","Stratified Lie groups: free boundary solutions exist","Mountain pass route to free boundary on stratified groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the quoted monotonicity lemma giving uniform horizontal gradient estimates, and on the standing assumption that the approximating mountain pass points u_j are bounded in $W^{{1,2}}$_0(Ω) ∩ L^∞(Ω); if either premise fails, the compactness that produces the limit solution is no longer available.","fun_headline_variants_meta":{"raw":{"variants":["Free boundary solutions now proven on Lie groups","Non-Euclidean free boundary problem solved","Stratified Lie groups: free boundary solutions exist","Mountain pass route to free boundary on stratified groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001445,"raw_usage":{"total_tokens":5731,"prompt_tokens":766,"completion_tokens":4965,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":382,"completion_tokens_details":{"reasoning_tokens":4906}},"tokens_in":382,"tokens_out":4965,"duration_ms":30740,"temperature":1.0,"reasoning_tokens":4906,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:57:21.296827+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct, on the unit ball of a stratified Lie group, a sequence u_j of Lipschitz functions satisfying the distributional inequalities of Lemma 3.1 with ε_j → 0 but with |∇_G u_j| unbounded; such an example would invalidate the monotonicity estimate that the proof uses. Alternatively, exhibit a bounded domain and a nonlinearity g obeying (g1)–(g2) for which the mountain pass critical points of E_ε are not uniformly bounded in L^∞ as ε → 0.","supporting_citations":[{"cited_title":"Choudhuria, D.D","cited_arxiv_id":null,"evidence_quote":"Supplies the monotonicity lemma in the stratified Lie group setting and treats the power-nonlinearity case that this paper generalizes."},{"cited_title":"Caffarelli, D","cited_arxiv_id":null,"evidence_quote":"Contains the classical Euclidean monotonicity theorem underlying the non-Euclidean version used here."}],"review_version":1}