{"id":"7736b6a7-a14a-4bcd-b0c8-a29fe94283a4","arxiv_id":"2506.00392","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The vector-valued Allen-Cahn equation with Robin boundary conditions converges locally in time to mean curvature flow with contact angle, and the limits solve harmonic heat flow and a minimal pair condition.","lead":"This paper proves that solutions of a vector-valued Allen-Cahn equation with Robin boundary conditions converge locally in time to mean curvature flow with a fixed contact angle at the boundary. It extends prior rigorous results for the scalar case with boundary contact and for vector-valued equations without boundary effects.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sign error in the relative-entropy functional B (Eq. 3.3): with ϑ<0 in Ω+ and ψ≤cF, the integrand (cFχΩ+−ψ)ϑ is non-positive, so (3.35a) does not control |B| and the proof of the L1 rate (1.18c) is incomplete as written.","rationale":"The reader identified the imported calibration lemma (Lemma 2.9) as the weakest assumption. That is a legitimate external-input concern, but it is a standard citation to a published result and the paper explicitly records the 2D restriction it entails. In contrast, the sign inconsistency in the definition of B is internal to this paper and directly affects the derivation of the quantitative phase-indicator estimate (1.18c), which is one of the two central quantitative claims of Theorem 1.1. Since B as written is always non-positive, the claimed bound (3.35a) is vacuous, and the slicing step (3.48) applied to g=ψ−cFχΩ+ requires control of −B, not B. This is a load-bearing gap in the written proof. The gap appears to be a sign typo rather than a substantive mathematical failure: replacing B by its absolute value or changing the sign in (3.3) makes the Gronwall argument and the subsequent slicing estimate valid. Therefore the appropriate disposition is CONDITIONAL: the central theorems are plausible and likely correct after a notational sign correction, but the manuscript as written does not fully prove (1.18c). I do not see a need to reject the paper, and I partially agree with the reader's conditional verdict, though for a different and more concrete reason than the calibration lemma.","tokens_in":41538,"tokens_out":24347,"duration_ms":245349,"concrete_test":"Take the well-prepared initial data from Appendix B and a fixed time t; on Ω+ near the interface cF−ψ>0 and ϑ<0, while on Ω−, −ψ<0 and ϑ>0, so the integrand (cFχΩ+−ψ)ϑ is pointwise non-positive and B<0 for any nontrivial profile. Then re-run Corollary 3.5 with B replaced by −B (equivalently ∫(ψ−cFχΩ+)ϑ); verify that the differential inequality becomes d/dt(−B) ≤ Cε + C(−B) and Gronwall yields |B(t)|≤Cε, which is the estimate actually needed for (3.35b). If this check succeeds, the mathematical theorem survives but the manuscript must correct the sign in (3.3) and all subsequent uses of B.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The functional Bε[u|Γ] defined in (3.3) is asserted to satisfy sup B ≤ C1ε and is then used in (3.35b) to conclude sup ∫|ψ−cFχΩ+| ≤ C1ε^{1/2}. However, under the stated sign convention (2.42a)-(2.42b), ϑ<0 in the interior of Ω+ and ϑ>0 outside Ω+. Since ψ=dF(u) takes values in [0,cF], the integrand (cFχΩ+−ψ)ϑ is ≤0 everywhere: in Ω+ we have cF−ψ≥0 and ϑ<0; in Ω− we have −ψ≤0 and ϑ>0. Hence Bε[u|Γ](t)≤0 for all u. The estimate (3.35a) is therefore vacuous as an upper bound on B, and it cannot control the quantity ∫|ψ−cFχΩ+||ϑ|, which equals −B. The Gronwall argument in Corollary 3.5, Step 1, bounds d/dt B above by Cε+CB; with B≤0 this only gives an upper bound on a non-positive quantity and does not prevent B from being very negative. Thus (3.35b) does not follow from the written estimates. The intended argument is recoverable by defining B with the opposite sign, e.g. B:=∫(ψ−cFχΩ+)ϑ (or by replacing B with |B| throughout), in which case the same estimates yield |B|≤Cε. This is a genuine internal inconsistency in the written proof of the central convergence rate, though one that appears fixable by a sign correction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the vector-valued Allen-Cahn equation with a double-well potential vanishing on two compact submanifolds N± ⊂ R^k, subject to Robin boundary conditions with a boundary energy density σ. The main results assert that, for well-prepared initial data satisfying (1.17), the relative entropy E_ε[u_ε|Γ] and the auxiliary functional B_ε[u_ε|Γ] grow at most O(ε), yielding an O(ε^{1/2}) L1-convergence rate of d_F(u_ε) to c_F χ_{Ω+_t} (Theorem 1.1); subsequential limits then solve harmonic heat flow into N± in the bulk and satisfy the minimal pair condition on the interface (Theorem 1.2). The proof combines the relative entropy method with boundary-adapted gradient flow calibrations from Hensel and Moser, and uses SBV regularity and geometric measure theory to treat boundary terms.","tokens_in":41868,"tokens_out":12195,"duration_ms":128784,"significance":"If the results are correct, the paper makes a valuable contribution by extending the vector-valued Allen-Cahn sharp-interface theory to domains with boundary contact energy, generalizing both the vector-valued interior result of Liu and the scalar-valued Robin result of Hensel and Moser. The technical machinery is appropriate: relative entropy estimates, gradient flow calibrations, and SBV compactness are used in a coherent way, and the paper is explicit about the two-dimensional restriction stemming from the calibration lemma. The construction of well-prepared initial data in Appendix B is also a useful and nontrivial part of the contribution. The main caveat is that the proof of the central convergence rate has a sign inconsistency in the auxiliary functional B, as detailed below.","major_comments":[{"comment":"The functional B is defined in (3.3) with a sign convention that makes it identically non-positive for every admissible u_ε. Indeed, by (2.42a)-(2.42b), ϑ<0 in the interior of Ω+_t and ϑ>0 outside Ω+_t, while ψ_ε=d_F(u_ε) takes values in [0,c_F]; hence (c_F χ_{Ω+_t}−ψ_ε)ϑ ≤ 0 pointwise in all of Ω. Therefore sup_t B_ε≤C_1 ε in (3.35a) is vacuous and cannot be used to control the quantity ∫|ψ_ε−c_F χ_{Ω+_t}||ϑ| dx, which equals −B_ε. The Gronwall argument in Step 1 bounds d/dt B_ε above by Cε+CB_ε, which only gives an upper bound on a non-positive quantity and does not prevent B_ε from becoming very negative. Consequently, (3.35b) and hence the L1 rate (1.18c) do not follow from the written estimates. The argument is recoverable by defining B with the opposite sign, i.e. B_ε := ∫(ψ_ε−c_F χ_{Ω+_t})ϑ dx, or equivalently by estimating −B_ε throughout; with that change the same estimates yield sup |B_ε|≤C_1 ε. As written, however, this is a load-bearing gap in the proof of Theorem 1.1.","section":"§3.1, Eq. (3.3); §3.2, Corollary 3.5"},{"comment":"The slicing inequality asserted in (3.47)-(3.48) is not valid for arbitrary g ∈ L∞, as stated. For example, taking g to be the indicator function of a thin tube of width w and length L around a curve gives (∫|g|)^2 of order L^2w^2 while ∫|g||ϑ| is of order Lw^2, so the ratio is of order L and can be made arbitrarily large by choosing a sufficiently long, thin support, even with |g| bounded. The proof of the claim uses only the boundedness of g and the geometric decomposition, so it does not justify the step from (3.35a) to (3.35b). If the inequality is only intended for g with special structure, such as the diffuse-interface profile ψ_ε−c_F χ_{Ω+_t} whose support has width controlled by ε, that structure must be stated and exploited explicitly.","section":"§3.2, Corollary 3.5, Step 2"}],"minor_comments":[{"comment":"Lemma 2.6 is foundational for the paper, but Appendix A provides only an outline and refers to [19, Appendix A] for the full argument. Since the paper relies on this lemma for existence, uniqueness, L∞-bounds, regularity, and the energy dissipation identity, please either include a complete proof or state precisely which statements in [19, Appendix A] apply verbatim to the current setting.","section":"Appendix A"},{"comment":"In the Notations section, the phrase \"symmetric differccne\" should read \"symmetric difference\".","section":"§1.4"},{"comment":"In equation (2.3), the density computation implicitly uses that E∪F agrees with Ω up to measure zero in the limit; since this is a standard fact for complementary sets in a bounded domain, a short clarifying sentence would improve readability.","section":"§2.1, Lemma 2.1"}],"recommendation":"major_revision","confidential_remarks":"The sign error in the definition of B appears to be a genuine but local mistake: the overall relative-entropy framework, the energy estimates, and the SBV-based regularity argument are well organized, and the proof is likely repairable by tracking −B instead of B. Given the paper's otherwise solid structure and the acknowledged dependence on the calibration lemma from [19], I would encourage a revision rather than rejection, provided the authors fix the sign inconsistency and clarify the slicing inequality in Corollary 3.5."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this is a genuine new theorem: the first vector-valued Allen-Cahn convergence result with Robin boundary conditions and high-dimensional wells, extending Liu's interior result and Hensel-Moser's scalar boundary result. The relative-entropy machinery is adapted carefully, the SBV regularity upgrade is non-trivial, and the restriction to 2D is stated honestly as a consequence of the calibration input. Second, there is a real sign error in the proof of the L1 convergence rate (1.18c). The functional B defined in (3.3) is non-positive under the stated sign convention (2.42a)-(2.42b), so the estimate sup B ≤ C1ε is vacuous and cannot control ∫|ψ - c_Fχ_Ω+||ϑ|, which equals -B. The Gronwall step only bounds an upper bound of a non-positive quantity. This is a load-bearing flaw in the written proof, but it looks fixable by flipping the sign of B (or running the argument for |B|); the rest of the relative entropy estimates appear coherent. The proof of Lemma 2.6 is only sketched in Appendix A, and Lemma 2.9 is imported from [19] without proof. Those are softer issues: Lemma 2.6 is standard parabolic regularity, and Lemma 2.9 is a known external input, though the paper would be stronger with a full proof or a precise reference. The minimal pair and harmonic heat flow sections are well done, with the BV/SBV compactness argument spelled out. This paper deserves a serious referee. I would send it to review, but I would flag the sign issue as a required revision. The author clearly knows the literature and the strategy is not circular; it uses external results as inputs. I'd cite this once the sign issue is fixed.","headline":"Genuine new theorem for vector-valued Allen-Cahn under Robin boundary conditions, but the proof of the central L1 rate has a sign error in the relative-entropy functional B; fixable and otherwise sound.","tokens_in":42362,"tokens_out":3767,"would_cite":true,"duration_ms":36842,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B25","35K55","53E10","58E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For vector-valued Allen-Cahn with boundary contact energy, well-prepared initial data yield convergence to mean curvature flow with fixed contact angle, at rates $\\epsilon$ in relative entropy and $\\epsilon^{1/2}$ in phase-field error…","keywords":["vector-valued Allen-Cahn","Robin boundary conditions","mean curvature flow","contact angle","relative entropy","harmonic heat flow","minimal pair condition","high-dimensional double-well potentials"],"falsifier":"Pick a smooth two-dimensional domain and a boundary energy for which planar mean curvature flow with contact angle $\\alpha\\in(0,90^\\circ]$ develops a singularity or pinch before time $T$, and check whether any triple $(\\xi,H,\\vartheta)$ can satisfy (2.41a)-(2.42f) up to that time; failure of the calibration would make the relative entropy inequality (3.12) impossible and would falsify Theorem 1.1 for that configuration. Alternatively, simulate the vector-valued Allen-Cahn equation (1.3) with a simple double-well pair such as two spheres in $\\mathbb{R}^3$ and measure $\\int_\\Omega |d_F(u_\\epsilon)-c_F\\chi_{\\Omega^+_t}|\\,dx$; a decay rate strictly slower than $\\epsilon^{1/2}$ would contradict (1.18c).","tokens_in":41312,"feed_emoji":"📐","tokens_out":7773,"duration_ms":73252,"temperature":0.7,"pith_summary":"This paper claims that, for a vector-valued Allen-Cahn equation with a high-dimensional double-well potential and nonlinear Robin boundary conditions, well-prepared initial data produce solutions whose diffuse interface converges to a sharp mean curvature flow meeting the boundary at a fixed contact angle $0<\\alpha\\le 90^\\circ$. The convergence is quantitative: the relative entropy stays of order $\\epsilon$ and the phase-field error $\\int_\\Omega |d_F(u_\\epsilon)-c_F\\chi_{\\Omega^+_t}|\\,dx$ decays like $\\epsilon^{1/2}$, uniformly on a time interval $[0,T]$. In the limit, the two phase fields take values in the well manifolds $N_\\pm$ in the bulk, evolve by harmonic heat flow into those manifolds, and on the interface satisfy the minimal pair condition $|u_+-u_-|=\\mathrm{dist}_N$. A sympathetic reader should care because this combines the interior vector-valued high-well setting with boundary contact energy and a general fixed contact angle, which is the natural regime for wetting and multi-component phase transitions.","feed_headline":"Vector-valued Allen-Cahn converges to sharp interface at rate √ε","feed_subtitle":"Boundary contact energy included: the limiting interface obeys mean curvature flow and Young's angle law.","key_machinery":"The argument is carried by the relative entropy functionals $E_\\epsilon[u_\\epsilon|\\Gamma]$ and $B_\\epsilon[u_\\epsilon|\\Gamma]$, built from the quasi-distance $d_F$ (a phase-field potential that is $0$ on $N_-$ and $c_F$ on $N_+$), together with a boundary-adapted gradient flow calibration triple $(\\xi,H,\\vartheta)$ supplied by Lemma 2.9. The calibration provides an extension of the interface normal, velocity, and signed distance with error estimates near the interface and boundary conditions $\\xi\\cdot n_{\\partial\\Omega}=\\cos\\alpha$, $H\\cdot n_{\\partial\\Omega}=0$; these enter the relative entropy inequality (3.12), whose Gronwall argument yields the $\\epsilon$ and $\\epsilon^{1/2}$ rates. To upgrade the weak limits $u_\\pm$ so that they are well-defined on the interface, the proof uses $SBV$ compactness and a geometric inequality comparing $|\\nabla u|$ with the gradient of its nearest-point projection onto the well manifolds, and a contradiction argument with Lemma 4.8 forces the minimal pair condition.","core_discovery":"Theorem 1.1 is the central assertion: for any strong mean-curvature-flow solution on a smooth two-dimensional domain with fixed contact angle, and for initial data satisfying (1.17), the unique weak solution satisfies $\\sup_t E_\\epsilon[u_\\epsilon|\\Gamma](t)\\le C_1\\epsilon$, $\\sup_t B_\\epsilon[u_\\epsilon|\\Gamma](t)\\le C_1\\epsilon$, and $\\sup_t\\int_\\Omega|d_F(u_\\epsilon)-c_F\\chi_{\\Omega^+_t}|\\,dx\\le C_1\\epsilon^{1/2}$. Theorem 1.2 then identifies the limits $u_\\pm$ as solutions of harmonic heat flow into $N_\\pm$ in the bulk, and shows the minimal pair condition holds $\\mathcal{H}^1$-a.e. on the interface. The paper thereby establishes the sharp-interface limit and the limiting system for the vector-valued Allen-Cahn equation with Robin boundary conditions, for contact angles up to and including $90^\\circ$.","pith_inferences":["A natural next test is whether the $O(\\epsilon^{1/2})$ phase-error bound is sharp; the slicing argument used to derive it suggests it is a width effect, so numerical experiments should compare against $C\\epsilon^{1/2}$ rather than expect a better exponent generically.","The calibration assumption of Lemma 2.9 is the true bottleneck for higher dimensions; constructing such triples in $\\mathbb{R}^3$ would extend Theorem 1.1 verbatim, and failure examples would identify exactly where the relative entropy machinery breaks.","The boundary coercivity condition $\\sigma(u)\\ge d_F(u)\\cos\\alpha$ is likely essential for the relative-entropy coercivity estimates; relaxing it would require a different control on the boundary term in (3.10a).","The minimal pair condition could be probed numerically by examining whether interfaces select a particular pair $(u_+,u_-)$ in the well manifolds as $\\epsilon\\to 0$, independently of the boundary energy."],"forward_implications":["If Theorem 1.1 is correct, the diffuse interface of the vector-valued Allen-Cahn equation with Robin boundary conditions converges, locally in time, to sharp mean curvature flow with fixed contact angle, with an explicit $O(\\epsilon^{1/2})$ bound on the bulk phase error.","The limiting phases $u_\\pm$ take values in the manifolds $N_\\pm$ and solve harmonic heat flow in the bulk, so the dynamics away from the interface are governed by the geometry of the well manifolds.","The minimal pair condition $|u_+-u_-|=\\mathrm{dist}_N$ holds on the interface, which selects admissible phase-boundary values in the vector-valued case.","The result covers the full range of contact angles $0<\\alpha\\le 90^\\circ$ and a broad class of boundary energy densities satisfying $\\sigma\\ge d_F\\cos\\alpha$ and Young's law, including the homogeneous Neumann case $\\alpha=90^\\circ$.","These conclusions generalize the scalar boundary-contact result and the interior vector-valued result to the combined vector-valued-with-boundary setting, and the same strategy applies to any calibration triple supplied by Lemma 2.9."],"supporting_citations":[{"why":"Supplies the boundary-adapted gradient flow calibration triple and the scalar boundary-contact baseline on which Lemma 2.9 rests.","marker":"[19]"},{"why":"Provides the vector-valued Allen-Cahn relative entropy framework, the quasi-distance function, and the minimal pair strategy without boundary effects.","marker":"[32]"},{"why":"Introduces the relative entropy method and the slicing inequality used to obtain the $\\epsilon^{1/2}$ phase error estimate.","marker":"[14]"},{"why":"Establishes the minimal pair condition for minimizers with high-dimensional double-well potentials, which the paper adapts to the dynamic setting.","marker":"[28]"},{"why":"Supplies the parabolic regularity and energy concentration lemmas for harmonic heat flow used to derive the limiting bulk equations.","marker":"[8]"},{"why":"Provides the SBV compactness, reduced boundary, and geometric measure theory tools used to upgrade the regularity of the limits $u_\\pm$.","marker":"[5]"},{"why":"Introduces gradient flow calibrations for weak-strong uniqueness, the concept adapted here to boundary contact phenomena.","marker":"[12]"}],"fun_headline_variants":["Vector Allen-Cahn: sharp interface limit with Robin BCs","Vector Allen-Cahn converges to mean curvature flow","Sharp interface limit for vector-valued Allen-Cahn with Robin BCs","Vector Allen-Cahn: Robin boundaries, mean curvature flow limit","Contact angle preserved in vector Allen-Cahn sharp-interface limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on Lemma 2.9, imported from reference [19], which asserts the existence of a strong planar mean curvature flow with contact angle together with a calibration triple satisfying estimates (2.41a)-(2.42f); the paper does not prove this lemma, and if such a calibration cannot be constructed for a given domain, interface, or boundary energy, the relative entropy inequality (3.12) and the convergence conclusions do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Vector Allen-Cahn: sharp interface limit with Robin BCs","Vector Allen-Cahn converges to mean curvature flow","Sharp interface limit for vector-valued Allen-Cahn with Robin BCs","Vector Allen-Cahn: Robin boundaries, mean curvature flow limit","Contact angle preserved in vector Allen-Cahn sharp-interface limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000851,"raw_usage":{"total_tokens":3676,"prompt_tokens":897,"completion_tokens":2779,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":2695}},"tokens_in":513,"tokens_out":2779,"duration_ms":20046,"temperature":1.0,"reasoning_tokens":2695,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:05:32.761431+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a smooth two-dimensional domain and a boundary energy for which planar mean curvature flow with contact angle $\\alpha\\in(0,90^\\circ]$ develops a singularity or pinch before time $T$, and check whether any triple $(\\xi,H,\\vartheta)$ can satisfy (2.41a)-(2.42f) up to that time; failure of the calibration would make the relative entropy inequality (3.12) impossible and would falsify Theorem 1.1 for that configuration. Alternatively, simulate the vector-valued Allen-Cahn equation (1.3) with a simple double-well pair such as two spheres in $\\mathbb{R}^3$ and measure $\\int_\\Omega |d_F(u_\\epsilon)-c_F\\chi_{\\Omega^+_t}|\\,dx$; a decay rate strictly slower than $\\epsilon^{1/2}$ would contradict (1.18c).","supporting_citations":[{"cited_title":"Hensel and M","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary-adapted gradient flow calibration triple and the scalar boundary-contact baseline on which Lemma 2.9 rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the vector-valued Allen-Cahn relative entropy framework, the quasi-distance function, and the minimal pair strategy without boundary effects."},{"cited_title":"Fischer, T","cited_arxiv_id":null,"evidence_quote":"Introduces the relative entropy method and the slicing inequality used to obtain the $\\epsilon^{1/2}$ phase error estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the minimal pair condition for minimizers with high-dimensional double-well potentials, which the paper adapts to the dynamic setting."},{"cited_title":"Chen and M","cited_arxiv_id":null,"evidence_quote":"Supplies the parabolic regularity and energy concentration lemmas for harmonic heat flow used to derive the limiting bulk equations."},{"cited_title":"Ambrosio, N","cited_arxiv_id":null,"evidence_quote":"Provides the SBV compactness, reduced boundary, and geometric measure theory tools used to upgrade the regularity of the limits $u_\\pm$."},{"cited_title":"Fischer and S","cited_arxiv_id":null,"evidence_quote":"Introduces gradient flow calibrations for weak-strong uniqueness, the concept adapted here to boundary contact phenomena."}],"review_version":1}