{"id":"c0af77ba-ced6-41ce-8868-d2bb3e1c5706","arxiv_id":"2501.03455","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Global weak solutions to volume-preserving mean curvature flow with obstacles exist in all dimensions by Allen-Cahn phase-field approximation.","lead":"This paper proves that volume-preserving mean curvature flow, a geometric motion that keeps the enclosed volume fixed, still has a well-defined weak solution when fixed obstacles are present. It works in all dimensions and uses a standard phase-field approximation, giving the first existence result for this combined problem.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Circularity/deferred proof of (5.19) in Prop. 8 and Thm. 2 is load-bearing: density-ratio bound and integrality of µt rest on it, and the text explicitly flags the circularity without resolving it.","rationale":"The reader's weakest_assumption coincides with the most load-bearing risk: the proof of Proposition 8 depends on (5.19), whose derivation is deferred and explicitly flagged as circular. I agree that this is the right center of gravity. The rest of the paper gives credible estimates: Proposition 5's L2 bound on λε via the elliptic test-function argument is reproduced in sufficient detail; the barrier construction in Lemma 1 handles the spatially dependent forcing; the proof of Part (D) is a standard phase-field-to-varifold passage once Theorem 2 holds. However, every one of those later steps ultimately uses the density-ratio bound or |ξ| = 0. A missing or subtly invalid adaptation of [24, Lemma 6.7] would therefore not just affect one numerical estimate; it would undermine the integrality (Theorem 4), the generalized mean curvature identification (Theorem 3), and the motion law (D). The paper's own inserted note about the circularity makes this the weakest link. No actual error was found in the displayed estimates; the concern is about an omitted proof. Hence the verdict should remain CONDITIONAL (equivalently, UNCHANGED from the reader's CONDITIONAL): the claim is credible but not fully verified until (5.19) is supplied for the obstacle forcing.","tokens_in":25522,"tokens_out":6913,"duration_ms":61883,"concrete_test":"Independently derive (5.19) for gε of (1.10), following [24, Lemma 6.7] and using only (5.7), Proposition 7, and the assumed bound supt∈[0,t̃] Dε(t) ≤ D1, without invoking Proposition 8. Check (i) whether the positive-part discrepancy is controlled on the whole space-time with the given forcing, including the transition layer across ∂O where gε is nonsmooth; (ii) whether the constant c'' can be chosen depending only on D1, T, d, R0, α; and (iii) whether inserting it into the contradiction algebra of Prop. 8 (with D1 = C1 + 1) yields an inequality that fails for all sufficiently small ε, as claimed. If (i)–(iii) hold, the circularity is benign; otherwise the density-ratio bound and the proof of |ξ| = 0 are incomplete, and Theorem 1 as stated is unverified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5.2's Proposition 8, which supplies the ε-uniform density-ratio bound Dε(t) ≤ DT, is proved by contradiction using estimate (5.19). The paper does not derive (5.19); it says it follows from [24, Lemma 6.7] 'with necessary changes made in a straightforward manner using (5.7)', and immediately before the proof states that 'the relation between Proposition 8 and (5.19) appears to be a circularity', deferring to [24, Section 6, (6.4)]. This matters because (5.19) is also invoked in Lemma 7 and Theorem 2 (|ξ| = 0), which in turn are the input to Theorem 3 (rectifiability, generalized mean curvature) and Theorem 4 (integrality µt = θH^{d−1}), and to Part (D)'s motion law. If the [24] argument does not survive the spatially dependent, piecewise-defined forcing gε of (1.10) — for example, if it needs a maximum-principle nonpositivity of ξε that fails here (the paper states this is exactly the obstruction), or if the continuity argument breaking the circularity requires a boundary condition or a forcing regularity stronger than (5.7) — then the density bound, the vanishing of |ξ|, and hence Theorem 1 collapse. This gap is structural, not merely cosmetic: no proof of (5.19) is present, and the self-located note confirms the author's awareness of the circularity.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves global-in-time existence of weak solutions (L2-flows in the sense of Mugnai and Roger) to volume-preserving mean curvature flow with obstacles in all dimensions d >= 2, via the phase-field method. The Allen-Cahn equation is augmented with a spatially dependent forcing g_epsilon designed near the obstacles to prevent intrusion into O_+ and O_-, and with a nonlocal multiplier lambda_epsilon. The main theorem asserts convergence of the associated energy measures to an integral varifold mu_t, convergence of the phase fields to a BV characteristic function with preserved volume and obstacle non-overlap, and the motion law v = h - (lambda/theta) nu on the reduced boundary away from the obstacles. The proof follows the strategy of Takasao [48] and Kagaya [24], with new estimates for the spatially dependent forcing (Propositions 2, 5, 7) and a new density-ratio bound (Proposition 8).","tokens_in":1626,"tokens_out":4142,"duration_ms":77791,"significance":"If correct, this is the first existence result for the combined volume-preserving mean curvature flow with obstacles, and it extends the phase-field approach to a setting where the maximum principle is unavailable and the discrepancy measure does not satisfy the usual nonpositivity. The paper is clearly written and identifies the main technical difficulty - the spatially dependent forcing and the loss of nonpositivity of xi_epsilon - and introduces Propositions 7 and 8 as a way to control the extra term in the monotonicity formula. However, the proof currently rests on several substantial deferred arguments, most importantly the estimate (5.19), and the paper explicitly acknowledges an apparent circularity between Proposition 8 and (5.19). The contribution would be solid once these gaps are filled.","major_comments":[{"comment":"The proof of the epsilon-uniform density-ratio bound D_epsilon(t) <= D_T is by contradiction and relies on estimate (5.19), which is asserted to follow from [24, Lemma 6.7] 'with necessary changes made in a straightforward manner using (5.7)'. Immediately before Proposition 8 the text states that 'the relation between Proposition 8 and (5.19) appears to be a circularity' and refers to [24, Section 6, (6.4)] for clarification, but the resolution is not reproduced. This is load-bearing because (5.19) is subsequently used in Lemma 7, Theorem 2, and Theorem 4; without it, the density bound, the vanishing of |xi|, and the integrality of mu_t would not follow. The manuscript must either prove (5.19) directly in the present spatially dependent setting or include the full circularity-breaking bootstrap argument from [24] adapted to the forcing g_epsilon.","section":"§5.2, Proposition 8 and (5.19)"},{"comment":"The proof of the vanishing of the discrepancy measure is incomplete: after deriving a finiteness condition, the text says 'We omit the rest of the argument as it is similar to the proof of [48, Theorem 13]' and only indicates changes to equation [48, (78)]. Since Theorem 2 is the key input to rectifiability (Theorem 3), integrality (Theorem 4), and part (D) of Theorem 1, and since the nonpositivity of xi_epsilon fails in this setting, the omitted part is not a routine repetition of [48]; the modifications should be written out fully.","section":"§5.4, Theorem 2"},{"comment":"The integrality theorem, which yields part (A.iii) of Theorem 1, is stated without proof. The text says the arguments of [48, Subsection 4.4] hold with 'necessary modifications' to [48, (96), (122)] and the last line of [48, p. 40], using Proposition 6, (5.19), and Theorem 2. Given the dependence on the unproved (5.19) and on Theorem 2, the reader cannot currently verify integrality. These modifications should be presented explicitly, especially where the nonconstant forcing g_epsilon enters.","section":"§5.5, Theorem 4"}],"minor_comments":[{"comment":"The phrase 'mean curvature flow with in the presence of obstacles' should read 'mean curvature flow in the presence of obstacles'.","section":"Abstract"},{"comment":"In the definition of the first variation, the test vector field is introduced as zeta in C^1_c(Omega; R^d), but the integral uses a different symbol; a single symbol should be used for the test field.","section":"§1.4, first variation definition"},{"comment":"The display contains 'g_epsilon dot (k(phi_epsilon(t)) - k(phi_epsilon(0)) dx', which is missing a closing parenthesis; it should be 'g_epsilon (k(phi_epsilon(t)) - k(phi_epsilon(0))) dx'.","section":"§2.2, display (2.7)"},{"comment":"The constant C_2 appearing in the final inequality is not defined; it should presumably be C_1, the contradiction constant introduced at the start of the proof.","section":"§5.2, proof of Proposition 8"},{"comment":"There are several typographical slips: 'Helley's selection theorem' should be 'Helly's selection theorem' (p. 25), and 'seel' should be 'see' (p. 12).","section":"Various"},{"comment":"The empty set is sometimes denoted by phi (e.g., 'M0 cap partial O = phi' on p. 4); use 'emptyset' or similar to avoid confusion with the phase field phi_epsilon.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central problem is well chosen and the strategy is believable, but it is atypical for this journal to have so many centrally placed results deferred to other papers with phrases like 'similar to [48]' and 'in a straightforward manner'. In particular, the apparent circularity between Proposition 8 and (5.19) is acknowledged in the text and left to [24, Section 6]; I would ask the author to give a complete, self-contained proof of (5.19) or to explicitly state and prove the circularity-breaking continuity argument in the present spatially dependent setting. The paper may also benefit from moving some of the 'identical to [48]' proofs to an appendix, or at least stating the precise changes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this about the paper: it proves the first existence of global weak solutions to volume-preserving mean curvature flow with obstacles, in all dimensions, via a phase field approximation. It is a real result, and the main strategy is credible. The author constructs a spatially dependent forcing g_epsilon that combines the volume penalty multiplier lambda_epsilon with an obstacle forcing, and shows that the obstacle forcing wins over the multiplier because |lambda_epsilon| decays like C epsilon^{-alpha/2}. The barrier argument (Lemma 1) and the L^2 bound on lambda_epsilon (Proposition 5) are the heart of the paper, and both are genuinely new and reasonably executed.\n\nWhat is new and good: the choice of multiplier (1.9) and forcing (1.10) is tailored to the obstacle geometry, and the comparison argument is original. The proof of Proposition 5 using an elliptic Neumann problem on Omega\\O and the mollified volume difference is clever. Propositions 7 and 8 (upper bounds on the discrepancy and density ratio) are adaptations of known techniques from Kagaya and Nik-Takasao, but the extension to a spatially dependent, piecewise forcing is nontrivial.\n\nSoft spots: several load-bearing arguments are deferred. Theorem 2 ends with \"We omit the rest of the argument as it is similar to the proof of [48, Theorem 13].\" Theorem 4 (integrality) is stated without proof. Most seriously, the density-ratio bound (Proposition 8) relies on estimate (5.19), which is quoted from [24, Lemma 6.7] \"with necessary changes made in a straightforward manner using (5.7)\" — and the text immediately acknowledges that the relation between Proposition 8 and (5.19) \"appears to be a circularity\", referring to [24, Section 6, (6.4)]. That is exactly the place where the spatially dependent forcing, whose nonpositivity fails, could break the argument. A referee will need to see the details of (5.19) actually worked out, or a clear statement of why the continuity argument in [24] goes through here. Without it, the density bound, the vanishing |\\xi|, and therefore rectifiability and integrality all rest on a citation.\n\nThat said, I did not find a load-bearing error in what is proved. The paper is honest about its gaps, and the gaps are omissions rather than obvious mistakes. The main theorem is plausible and important. If the deferred arguments are supplied, this becomes a solid paper in the area.\n\nWho it is for: geometric analysts working on phase field methods, volume-preserving flows, and obstacle problems. It deserves a serious referee, and I would send it to review rather than desk reject. My own verdict would be conditional: accept if the author fully writes out the proof of (5.19), or at minimum demonstrates that the cited [24] argument survives the spatially dependent forcing. I would not cite it in my own work until those details are available.\n\nRecommendation: engage with it, but ask for the missing proof.","headline":"First existence result for volume-preserving mean curvature flow with obstacles, genuinely novel and credible, but a central density-ratio estimate is deferred to a cited reference and needs to be spelled out before the main theorem is fully verified.","tokens_in":26342,"tokens_out":3326,"would_cite":false,"duration_ms":29663,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","53E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that volume-preserving mean curvature flow constrained by fixed obstacles admits global weak solutions in every dimension, obtained as limits of an Allen-Cahn phase-field equation with a multiplier and obstacle forcing.","keywords":["volume-preserving mean curvature flow","obstacle problem","Allen-Cahn equation","varifolds","weak solutions","phase field method","discrepancy measure","L2-flow"],"falsifier":"Check whether estimate (5.19) can actually be derived for the forcing $g_\\varepsilon$ under the assumptions (5.7), following the continuity argument cited from [24, Section 6, (6.4)]. If the argument cannot be adapted, the density-ratio bound of Proposition 8 has no proof, and the rectifiability and integrality conclusions would be unsupported. A complementary numerical check: solve (1.8) with a fixed obstacle and monitor whether $\\sup_{t\\in[0,T]} D_\\varepsilon(t)$ remains bounded uniformly in $\\varepsilon$; unbounded growth would disprove Proposition 8 and the $(d-1)$-integrality clause.","tokens_in":25277,"feed_emoji":"🫧","tokens_out":19846,"duration_ms":159758,"temperature":0.7,"pith_summary":"Volume-preserving mean curvature flow moves an interface with normal velocity equal to its mean curvature minus a time-dependent multiplier chosen to keep the enclosed volume fixed. This paper proves that when the interface is additionally forbidden from entering fixed obstacles, a global weak solution still exists in every dimension $d\\ge 2$, by showing that solutions of a suitably modified Allen-Cahn phase-field equation converge to such a flow. The modification adds a spatially dependent forcing that pushes the interface away from the obstacles, and the main technical achievement is controlling this forcing against the nonlocal volume-preserving multiplier. If the result is correct, it supplies the first existence theorem for the combined obstacle-plus-volume-preserving problem and validates a concrete phase-field scheme for computing constrained interface motions such as those arising in cell motility models.","feed_headline":"Obstacle-blocked interfaces move while preserving volume","feed_subtitle":"An Allen-Cahn phase-field approximation converges to a sharp interface that respects obstacles and conserves volume in any dimension.","key_machinery":"The central object is the modified Allen-Cahn equation (1.8), $\\varepsilon\\partial_t\\phi_\\varepsilon = \\varepsilon\\Delta\\phi_\\varepsilon - \\varepsilon^{-1}W'(\\phi_\\varepsilon) + g_\\varepsilon\\sqrt{2W(\\phi_\\varepsilon)}$, with double-well potential $W(r)=\\tfrac12(1-r^2)^2$, multiplier $\\lambda_\\varepsilon(t)$ defined in (1.9) as an $\\varepsilon^{-\\alpha}$-normalized integral of $\\eta_\\varepsilon(s(x))(k(\\phi_\\varepsilon^0)-k(\\phi_\\varepsilon))$, and the spatially dependent forcing $g_\\varepsilon$ of (1.10) interpolating between $\\lambda_\\varepsilon\\eta_\\varepsilon$ in a $\\sqrt{\\varepsilon}$-neighborhood of the obstacles and $\\pm d/R_0$ on the obstacle interiors. This is the mechanism that converts volume preservation into a normal-velocity constraint while keeping the interface out of $O_\\pm$: the gradient-flow structure forces the bound $|\\lambda_\\varepsilon(t)|\\le C\\varepsilon^{-\\alpha/2}$, which is strong enough to make the obstacle forcing win in the comparison principle, to give the $L^2$ bound in time on $\\lambda_\\varepsilon$, and, through the monotonicity formula and the upper bounds on the discrepancy and density ratio, to yield the vanishing $|\\xi_\\varepsilon|\\to 0$ that underlies rectifiability and integrality.","core_discovery":"The paper's central claim is Theorem 1: from any $C^1$ initial surface $M_0=\\partial U_0$ separating two fixed disjoint obstacles $O_+\\subset U_0$ and $O_-\\subset\\Omega\\setminus U_0$ with $C^{2,\\beta}$ boundaries, there is a subsequence of solutions to the Allen-Cahn system (1.8) whose associated energy measures converge to a family of Radon measures $\\mu_t$ forming a global weak solution of the obstacle-constrained volume-preserving mean curvature flow in the L2-flow sense, meaning a family of rectifiable varifolds with an $L^2$ generalized mean curvature and a generalized velocity field. In the limit, $\\mu_t$ is a $(d-1)$-dimensional integral varifold for almost every time; the phase fields converge to the characteristic function of a finite-perimeter set $E(t)$ whose volume matches the initial volume and which contains $O_+$ and avoids $O_-$ for all time; and, away from the obstacles, the motion law $v = h - (\\lambda/\\theta)\\nu$ holds on the reduced boundary, with $\\lambda$ the weak $L^2_{\\mathrm{loc}}$ limit of the multipliers $\\lambda_\\varepsilon$. The constructive content is that the spatially dependent forcing can be handled: the estimate $|\\lambda_\\varepsilon(t)|\\le C\\varepsilon^{-\\alpha/2}$ lets the constant obstacle driving $\\pm d/R_0$ dominate the multiplier near $\\partial O$, the $L^2$ bound on $\\lambda_\\varepsilon$ yields the volume constraint in the limit, and the vanishing of the discrepancy measure $|\\xi_\\varepsilon|\\to 0$ provides the sharp-interface, integrality requirement.","pith_inferences":["A natural extension, not claimed by the paper, is to apply the same multiplier-versus-forcing comparison to other constrained flows (for example area-preserving curve shortening or surface diffusion with obstacles) whenever the conserved quantity admits an $\\varepsilon$-uniform bound of the type (2.9).","The $\\sqrt{\\varepsilon}$-cut-off layer in (1.10) suggests a testable size estimate: in the phase-field approximation, any penetration of the interface into the obstacles should be confined to a region whose thickness vanishes at most like $\\sqrt{\\varepsilon}$, which could be checked numerically.","If the circularity in the density-ratio argument cannot be resolved, a plausible alternative route to the theorem would be an original density estimate tailored to $g_\\varepsilon$ that does not pass through the borrowed lemma; the paper does not provide such a route.","The convergence results are subsequential and do not address uniqueness; a neighboring open problem is whether the limiting flow depends on the chosen subsequence or on the auxiliary parameters used to build the approximation."],"forward_implications":["A $C^1$ initial surface separating prescribed disjoint obstacles admits a global-in-time weak motion that conserves its enclosed volume and never penetrates the obstacles, in every spatial dimension $d\\ge 2$.","The Allen-Cahn system (1.8) with the multiplier (1.9) and forcing (1.10) is a convergent diffuse-interface approximation: its energy measures and phase fields pass to the sharp-interface limit with no volume drift and no obstacle overlap in the limit.","Away from the obstacles the limiting flow obeys the motion law $v = h - (\\lambda/\\theta)\\nu$, so the geometric content of volume-preserving mean curvature flow survives the transition to weak solutions.","The multiplier $\\lambda_\\varepsilon$ is uniformly bounded in $L^2_{\\mathrm{loc}}$ and passes to a weak limit $\\lambda$, giving a well-defined continuum Lagrange multiplier for the volume constraint.","The vanishing of the discrepancy measure means the limiting energy concentrates on a $(d-1)$-rectifiable integer-multiplicity surface, so the weak solution is genuinely an interface rather than a diffuse bulk effect."],"supporting_citations":[{"why":"Motivates the multiplier choice through the approximate volume-penalizing flow.","marker":"[25]"},{"why":"Supplies the weak-solution framework and the multiplier comparison that the construction adapts.","marker":"[41]"},{"why":"Provides the higher-dimensional phase-field convergence framework for volume-preserving mean curvature flow, including the estimate the paper extends to obstacles.","marker":"[48]"},{"why":"Gives the earlier two- and three-dimensional weak-convergence result and the monotonicity formula identity used in Section 5.","marker":"[46]"},{"why":"Defines the L2-flow notion of weak solution that Theorem 1 targets.","marker":"[40]"},{"why":"Supplies the Allen-Cahn convergence framework for varifold mean-curvature flow, the monotonicity formula, and the initial density bound.","marker":"[23]"},{"why":"Provides the obstacle barrier construction used to keep the interface out of the positive and negative obstacles.","marker":"[47]"},{"why":"Is the source of the density estimate and the circularity-resolving continuity argument the paper relies on without reproducing.","marker":"[24]"},{"why":"Supplies the forcing-term version of the density estimate invoked to make the borrowed argument available for the spatially dependent forcing.","marker":"[43]"}],"fun_headline_variants":["Weak solutions exist for obstacle-bounded volume-preserving flow","Volume-preserving flow with obstacles: weak solutions proven","Obstacle-aware mean curvature flow yields global weak solutions","Phase-field proof of obstacle-constrained volume-preserving flow","Obstacle-constrained volume-preserving flow: weak solutions exist"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a technical estimate the proof borrows from a cited argument — described in the paper only as following 'with necessary changes made in a straightforward manner' — really holds for this spatially variable forcing and is not secretly assuming the very conclusion it supports; the manuscript itself notes, before Proposition 8, that 'the relation between Proposition 8 and (5.19) appears to be a circularity' and defers the resolution to the cited source without reproducing it.","fun_headline_variants_meta":{"raw":{"variants":["Weak solutions exist for obstacle-bounded volume-preserving flow","Volume-preserving flow with obstacles: weak solutions proven","Obstacle-aware mean curvature flow yields global weak solutions","Phase-field proof of obstacle-constrained volume-preserving flow","Obstacle-constrained volume-preserving flow: weak solutions exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000605,"raw_usage":{"total_tokens":2858,"prompt_tokens":1017,"completion_tokens":1841,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":1759}},"tokens_in":633,"tokens_out":1841,"duration_ms":14310,"temperature":1.0,"reasoning_tokens":1759,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:52:31.196718+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether estimate (5.19) can actually be derived for the forcing $g_\\varepsilon$ under the assumptions (5.7), following the continuity argument cited from [24, Section 6, (6.4)]. If the argument cannot be adapted, the density-ratio bound of Proposition 8 has no proof, and the rectifiability and integrality conclusions would be unsupported. A complementary numerical check: solve (1.8) with a fixed obstacle and monitor whether $\\sup_{t\\in[0,T]} D_\\varepsilon(t)$ remains bounded uniformly in $\\varepsilon$; unbounded growth would disprove Proposition 8 and the $(d-1)$-integrality clause.","supporting_citations":[{"cited_title":"On obstacle problem for brakke’s mean curvature flow","cited_arxiv_id":null,"evidence_quote":"Provides the obstacle barrier construction used to keep the interface out of the positive and negative obstacles."},{"cited_title":"Volume preserving mean curvature flow for star-shaped sets","cited_arxiv_id":null,"evidence_quote":"Motivates the multiplier choice through the approximate volume-penalizing flow."},{"cited_title":"Global solutions to the volume- preserving mean-curvature flow","cited_arxiv_id":null,"evidence_quote":"Supplies the weak-solution framework and the multiplier comparison that the construction adapts."},{"cited_title":"The existence of a weak solution to volume preserving mean curvature flow in higher dimensions","cited_arxiv_id":null,"evidence_quote":"Provides the higher-dimensional phase-field convergence framework for volume-preserving mean curvature flow, including the estimate the paper extends to obstacles."},{"cited_title":"Existence of weak solution for volume preserving mean curvature flow via phase field method","cited_arxiv_id":null,"evidence_quote":"Gives the earlier two- and three-dimensional weak-convergence result and the monotonicity formula identity used in Section 5."},{"cited_title":"The allen-cahn action functional in higher dimensions","cited_arxiv_id":null,"evidence_quote":"Defines the L2-flow notion of weak solution that Theorem 1 targets."},{"cited_title":"Convergence of the allen-cahn equation to brakke’s motion by mean curvature","cited_arxiv_id":null,"evidence_quote":"Supplies the Allen-Cahn convergence framework for varifold mean-curvature flow, the monotonicity formula, and the initial density bound."},{"cited_title":"Convergence of the allen–cahn equation with a zero neumann boundary condition on non-convex domains","cited_arxiv_id":null,"evidence_quote":"Is the source of the density estimate and the circularity-resolving continuity argument the paper relies on without reproducing."},{"cited_title":"On an obstacle problem for the brakke flow with a gener- alized right-angle boundary condition","cited_arxiv_id":null,"evidence_quote":"Supplies the forcing-term version of the density estimate invoked to make the borrowed argument available for the spatially dependent forcing."}],"review_version":1}