{"id":"45abc1dd-6f8a-49e4-a5fb-e8d673f35322","arxiv_id":"2411.18491","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A mass-constrained phase-field energy for epitaxial growth with adatoms is shown to Gamma-converge to the sharp relaxed model with vertical fractures.","lead":"This mathematics paper proves that a smooth phase-field version of an epitaxial thin-film growth model converges, in a precise variational sense, to the sharp original model as a small parameter tends to zero. The model includes mobile surface atoms, called adatoms, and fixes the total film mass, which earlier phase-field treatments omitted.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The mass constraint is imposed on u, but the Gamma-limit tracks the measure mu_eps; this makes the Theorem 15 recovery sequence inadmissible for A_p(m,M) and destroys the asserted equi-coercivity, so Theorems 6 and 7 are not proven as stated.","rationale":"The reader's weakest assumption correctly identifies that bounded energy does not control mu_eps mass, so Theorem 5's convergence hypothesis is not derived and Theorem 7 lacks equi-coercivity. Our stress-test confirms this and goes further: the same mismatch between the constraint on u and the measure mu_eps also invalidates the recovery sequence in Theorem 15, Step 3, which is constructed to satisfy mu_eps(R^2)=m but never shown to satisfy the stated admissibility condition int u_eps = m. Thus the problem is not only in the minimizer-convergence corollary; the mass-constrained Gamma-convergence theorem itself (Theorem 6) lacks an admissible recovery sequence.\n\nWe do not read this as a fatal mathematical error. The construction in Theorem 15 computes mu_eps(R^2)=m by design, and Remark 6 shows the author is aware that support of the limit measure is not automatic. The natural fix is to change the constraint in Definition 19 to mu_eps(R^2)=m (or to require psi to grow so that int q_eps u_eps is controlled). With that correction, the recovery sequence becomes admissible, and the energy bound plus the fixed phase-field mass M and BV-bound h would yield the needed tightness. The mathematical core of the paper, combining [5] and [9], appears sound; what is missing is a consistent definition of the mass constraint and the resulting adjustments to Theorems 5-7.\n\nFor these reasons we agree with the CONDITIONAL verdict, but for a broader set of reasons than the reader gave: the concern is located in the formulation of the constrained problem, not merely in the compactness argument for minimizers. The paper should be revised to clarify the intended mass constraint and to either prove or assume the corresponding compactness statement. No issue is taken with the integrity of the exposition; the gaps are technical and localized.","tokens_in":22595,"tokens_out":17153,"duration_ms":151834,"concrete_test":"Verify the admissibility of the recovery sequence in Theorem 15, Step 3, in a flat-interface special case. Let Gamma = (a,b) x {0}, u == 1 so that m = H^1(Gamma), and choose a single grid rectangle R in a delta-cover with |R| < m. For the constructed sequence, u_eps = H^1(Gamma cap R)/p_eps on R, with p_eps := (1/sigma) int_R (eps|grad w_eps|^2 + (1/eps)P(w_eps)) dx. Since p_eps -> H^1(Gamma cap R) as eps -> 0, compute int_{Q+} u_eps dx = (H^1(Gamma cap R)/p_eps) |R| -> |R|, which is not equal to m. This directly falsifies (w_eps, v_eps, u_eps) in A_p(m,M). If instead the constraint is on mu_eps, the same computation confirms mu_eps(R^2) -> m, showing the constraint variable is the issue.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 19 imposes the constraint int_{Q+} u dx = m, while Definition 21(iii) defines the object that converges to the adatom measure as mu_eps = u_eps (1/sigma)(eps|grad w_eps|^2 + (1/eps)P(w_eps)) L^2 restricted to Q+. These are different quantities, and the mismatch is load-bearing.\n\nFirst, Theorem 5 assumes mu_eps weak-* converges to mu with supp mu subset Gamma, but this is not derived from the hypotheses. Since psi is only bounded below, sup F_eps < infinity yields a uniform L^1 bound on q_eps := (1/sigma)(eps|grad w_eps|^2 + (1/eps)P(w_eps)), not on int q_eps u_eps. Taking psi == 1 and u_eps = m/|A_eps| on the transition layer A_eps (with |A_eps| approx eps H^1(Gamma)) gives int u_eps = m, sup F_eps < infinity, but mu_eps(Q+) tends to infinity. Thus the family is not equi-coercive; the assertion before Theorem 7 that the family is equi-coercive is false.\n\nSecond, the recovery sequence in Theorem 15, Step 3, is constructed to satisfy mu_eps(R^2) = m, and the displayed computation verifies exactly that. It is never checked against the stated admissibility condition int_{Q+} u_eps dx = m. Since u_eps is constant on each grid rectangle R_j, int_{Q+} u_eps dx = sum_j u_j H^1(Gamma cap R_j)/p^j_eps |R_j|, which tends to sum_j u_j |R_j| as eps -> 0. In general this is not m; for a flat interface and a single small rectangle it can be made arbitrarily small. Hence (w_eps, v_eps, u_eps) is not in A_p(m,M) as defined, and the limsup inequality for Theorem 6 has no admissible recovery sequence.\n\nBoth symptoms point to a common root cause: the mass constraint is imposed on the wrong variable. If instead the constraint were mu_eps(R^2) = m, or if psi were given a growth condition, the compactness failure would disappear and the recovery sequence would be admissible. As stated, the central mass-constrained Gamma-convergence claim and the minimizer-convergence corollary are not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a phase-field approximation for a sharp model of epitaxial growth with adatoms previously relaxed in [9]. The phase-field functional couples a double-well Modica-Mortola surface term to an adatom density u through a coefficient ψ(u), and the main claimed results are Γ-convergence of the unconstrained functionals F_ε to F (Theorem 4), Γ-convergence of the constrained functionals F_ε^{m,M} to F^{m,M} (Theorem 6), and convergence of minima (Theorem 7), all with respect to the topology of Definition 21. The proof strategy adapts the methods of [3] and [5] to the relaxed functional of [9].","tokens_in":22974,"tokens_out":16799,"duration_ms":166336,"significance":"If the results were valid, the paper would provide a useful phase-field approximation of a nontrivial free-boundary model with adatoms, including a mass constraint, and would extend the approaches of Bonnetier-Chambolle and Caroccia-Cristoferi to the relaxed functional with cuts. The paper has no free parameters and relies on a previously published relaxation theorem rather than circular reasoning. However, the constrained version of the result is currently not established: the adatom mass constraint in Definition 19 is imposed on the unweighted L1 integral of u, while the object that passes to the limit is the energy-weighted measure μ_ε of Definition 21(iii). This mismatch breaks the compactness argument, the claimed equi-coercivity, and the admissibility of the recovery sequence.","major_comments":[{"comment":"The equality μ_ε(Q+)=m is not a consequence of the constraint ∫_{Q+}u_ε dx=m. By Definition 19 the constraint is on the unweighted L1 integral of u_ε, while μ_ε is the energy-weighted measure defined in Definition 21(iii). Since ψ is only assumed bounded below, the energy bound (10) does not control μ_ε(Q+). Taking ψ≡1 and u_ε=m/|A_ε| on the transition layer A_ε with |A_ε|≈εH^1(Γ) gives ∫_{Q+}u_ε dx=m and sup F_ε<∞, but μ_ε(Q+) tends to infinity; thus (13) is false even when the weak-* convergence in the hypotheses is assumed, because the limit of μ_ε(Q+) need not equal m. Consequently Theorem 5 does not establish that the limiting measure has mass m.","section":"§5, Eq. (13)"},{"comment":"The remark that the family (F_ε^{m,M})_ε is equi-coercive is false under the topology of Definition 21. The functional does not control the total mass of μ_ε: the example in the previous comment gives sup F_ε<∞ and ∫_{Q+}u_ε dx=m but no weak-* convergent subsequence of μ_ε. The same example shows that Remark 5 is incorrect, since strict positivity of ψ is not sufficient for coerciveness. A growth condition on ψ or a constraint imposed directly on μ_ε(R^2) would be needed before Theorem 7 can be derived from Theorems 2, 5 and 6.","section":"Before Theorem 7, Definition 2"},{"comment":"The recovery sequence is not checked against the admissibility condition in Definition 19. The computation verifies μ_ε(R^2)=m, but A_p(m,M) requires ∫_{Q+}u_ε dx=m. With the definition u_ε|_{R_j}=u_j H^1(Γ∩R_j)/p_j^ε on each grid rectangle R_j, one has ∫_{Q+}u_ε dx≈Σ_j u_j |R_j∩Q_+| (since p_j^ε→H^1(Γ∩R_j)), whereas m=Σ_j u_j H^1(Γ∩R_j). For a flat interface segment contained in a tall rectangle these two quantities differ in general. Hence the constructed (w_ε,v_ε,u_ε) is generally not in A_p(m,M), and the limsup inequality for Theorem 6 is not proven.","section":"§7, Theorem 15, Step 3 (Eqs. (51)–(54))"},{"comment":"The lower bound for the bulk term is not justified by the stated hypotheses. The sequence is only assumed to satisfy v_ε→v in L2_loc and w_ε→χ_Ω in L1_loc; no bound on E(v_ε) is available. The displayed argument 'by convexity and liminf w_ε=1' is not sufficient for the weak lower semicontinuity of the quadratic bulk energy. If the liminf is finite, one can extract a weighted L2 control on E(v_ε) on compact subsets of Ω and then use a truncation or Egorov argument, but this reasoning is absent from the proof. This gap affects the liminf inequality for Theorem 4 as well.","section":"§6, Theorem 10, Eq. (15)"}],"minor_comments":[{"comment":"There is a typo in the introduction: 'we condiderv∈H1(Ω;R2)' should read 'we consider v∈H1(Ω;R2)'.","section":"Abstract/Introduction"},{"comment":"The initial conditions γ_ε(0)=1, γ_ε(1)=0 in the almost optimal profile problem are stated without comment, while the earlier equation (9) uses the opposite convention. The sign convention is consistent with the subsequent extension γ_ε=1 for x<0 and γ_ε=0 for x>1, but the inconsistency should be resolved explicitly.","section":"§7, Eq. (33)"},{"comment":"In the proof of Theorem 10, the rectangle R_r is defined with the same x-interval as R_l; it should presumably be (x_c, x_c+δ) instead of (x_c-δ, x_c).","section":"§6, Step 2.1"},{"comment":"The sentence 'C_{ξγ} is monotonically converging to the empty set, as ε→0' should refer to γ→0, not ε→0.","section":"§6, Step 2"},{"comment":"The statement 'lim_{ε→∞} H(Ω_ε,v_ε,μ_ε)=...' should be lim_{ε→0}, and the reference 'where H is defined (14)' appears to point to the wrong equation, since (14) is the Γ-limit E from [5].","section":"§7, Theorem 14"}],"recommendation":"major_revision","confidential_remarks":"The main results of the constrained problem are not proven as stated, but the issues are localized and appear repairable: either change the mass constraint to a condition on μ_ε(R^2) or add a growth/compactness assumption on the adatom density, and correct the recovery sequence so that ∫_{Q+}u_ε dx=m. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it carries out a phase-field approximation for the sharp graph-based epitaxial model with adatoms from [9], adapting the cut-energy treatment and the wriggling process into a Gamma-convergence proof. The liminf argument, especially the localization around vertical cuts and the use of the convex sub-additive envelope, is detailed and mostly convincing. The unconstrained Gamma-convergence claim in Theorem 4 looks defensible, and the paper is honest about building on [3], [5], and [9]. No circularity, no data fitting.\n\nThe soft spot is real and load-bearing: the mass constraint is imposed on u, not on the measure that actually converges. Definition 19 requires ∫ u_ε dx = m, but the topology in Definition 21 tracks μ_ε = u_ε (1/σ)(ε|∇w|² + P(w)/ε) L². The recovery sequence in Theorem 15, Step 3, is engineered so that μ_ε(R²) = m; the computation verifies exactly that. It is never checked against ∫ u_ε dx = m. For a grid rectangle R_j, ∫_{R_j} u_ε dx ≈ u_j |R_j| / H¹(Γ∩R_j), which is not the sharp mass contribution and can be made arbitrarily small for a flat interface. So the sequence is not admissible for A_p(m,M), and Theorem 6's limsup has no admissible recovery sequence. Since Theorem 7 relies on Theorem 6 plus equi-coercivity, and equi-coercivity is also not established — with only inf ψ > 0, bounded energy does not control μ_ε mass — the constrained claims fail.\n\nThat said, the flaw is localized and fixable. The natural repair is to change the constraint to μ_ε(R²) = m, or add a growth condition on ψ that controls the μ mass, and then rebuild Step 3 to satisfy the actual admissibility condition. This is a serious technical gap, not a sign of a broken program. The author's self-assessment of what is new is accurate; the proof strategies are assembled, but the assembly is nontrivial.\n\nThis paper deserves a serious referee: the unconstrained result is likely recoverable, the gap is exactly the sort a careful referee should catch, and the author has been transparent about the missing link. I would not cite it as it stands, but I would engage with a revised version.","headline":"A plausible Gamma-convergence framework for phase-field epitaxy with adatoms, but the mass-constrained version breaks because the recovery sequence never satisfies the stated constraint on u.","tokens_in":770,"tokens_out":1779,"would_cite":false,"duration_ms":43972,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J45","74G65","49Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A smoothed phase-field energy is proved to converge to the sharp epitaxial-growth model with adatoms as the smoothing vanishes.","keywords":["phase-field approximation","Gamma-convergence","epitaxial growth","adatoms","linearised elasticity","free discontinuity problems","mass constraint","surface energy"],"falsifier":"Take $\\psi\\equiv1$ and put $u_\\varepsilon=m/\\varepsilon$ on the $\\varepsilon$-transition layer of $w_\\varepsilon$; the surface-energy term of $F_\\varepsilon$ stays bounded while $\\mu_\\varepsilon$ has mass of order $1/\\varepsilon$, so no subsequence can weak-* converge with support on the interface. This calculation makes concrete the failure flagged in Remark 6: without hypothesis (12), the compactness asserted in the equi-coercivity step behind Theorem 7 cannot hold.","tokens_in":22317,"feed_emoji":"⚛️","tokens_out":11219,"duration_ms":94890,"temperature":0.7,"pith_summary":"This paper proves a rigorous phase-field approximation, in the sense of $\\Gamma$-convergence, for the sharp variational model of epitaxial growth with adatoms studied in the author's companion paper [9]. The smooth functionals $F_\\varepsilon$ and their mass-constrained versions $F_\\varepsilon^{m,M}$ $\\Gamma$-converge, as $\\varepsilon\\to0$, to the sharp relaxed functionals $F$ and $F^{m,M}$: the phase-field interface collapses to the film boundary, the displacement converges in $L^2_{\\rm loc}$, and the weighted adatom measure converges weak-* to the limit measure. The payoff is that a numerically friendly model whose variables are all functions retains the exact surface energetics of the sharp model, including vertical fractures, the convex sub-additive envelope of the adatom cost, and the singular part of the adatom measure. As a direct corollary, the minimal energies of the constrained phase-field problems converge to the minimum of the sharp constrained problem, and minimizing sequences pass to sharp minimizers.","feed_headline":"Phase-field energy with adatoms converges to the sharp film model","feed_subtitle":"With mass constraints intact, the smooth minimizers recover minimizers of the sharp epitaxial-growth functional, fractures included.","key_machinery":"The carrying mechanism is the Modica–Mortola-type interfacial energy $\\frac1\\sigma\\int_{Q_+}(\\varepsilon|\\nabla w|^2+\\frac1\\varepsilon P(w))\\psi(u)\\,dx$, where $P$ is a double-well potential vanishing only at $0$ and $1$ and $\\sigma$ normalizes the one-dimensional profile. The double well forces $w$ to spend its energy in a transition layer of width $\\varepsilon$ that collapses onto the film boundary, and multiplying by $\\psi(u)$ turns that layer into the adatom measure $\\mu_\\varepsilon$; in the limit, the cost of a density $u$ on the regular interface is $\\widetilde\\psi(u)$, while vertical cuts are handled by splitting the layer left and right of the cut and combining the two one-sided costs through $\\psi^c$. Recovery sequences are built from the almost optimal profile $\\gamma_\\varepsilon$ solving $\\varepsilon^2|\\gamma_\\varepsilon'|^2=P(\\gamma_\\varepsilon)+\\sqrt\\varepsilon$ with $\\gamma_\\varepsilon(0)=1$, $\\gamma_\\varepsilon(1)=0$, composed with the signed distance to $\\Omega$; the 'wriggling process' from the earlier papers lets one approximate densities above the linear threshold of $\\widetilde\\psi$ by repeated copies of the threshold value $s_0$.","core_discovery":"On the paper's own terms, the discovery is that the phase-field functionals $$F_\\varepsilon(w,v,u)=\\int_{Q_+}(w+\\eta_\\varepsilon)W(E(v)-E_0(y))\\,dx+\\frac1\\$\\sigma$\\int_{Q_+}\\left(\\varepsilon|\\nabla w|^2+\\frac1\\varepsilon P(w)\\right)\\psi(u)\\,dx,$$ with $\\eta_\\varepsilon=o(\\varepsilon)$ and $\\sigma=2\\int_0^1\\sqrt{P(t)}\\,dt$, have $\\Gamma$-limit equal to the relaxed sharp energy $F$ from the earlier work, under the convergence of Definition 21: $w_\\varepsilon\\to\\chi_\\Omega$ in $L^1_{\\rm loc}$, $v_\\varepsilon\\to v$ in $L^2_{\\rm loc}$, and the measures $\\mu_\\varepsilon=\\frac{u_\\varepsilon}{\\sigma}(\\varepsilon|\\nabla w_\\varepsilon|^2+\\frac1\\varepsilon P(w_\\varepsilon))\\,\\mathcal L^2\\llcorner Q_+$ converge weak-* to $\\mu$. The limit separates the surface energy into three contributions: $\\widetilde\\psi(u)$ on the regular part $\\widetilde\\Gamma$, $\\psi^c(u)$ on the cut, fracture-like part $\\Gamma^c$, and $\\theta\\,\\mu_s$ on the singular part, where $\\widetilde\\psi$ is the convex sub-additive envelope of $\\psi$ and $\\psi^c(s)=\\min\\{\\widetilde\\psi(r)+\\widetilde\\psi(t): r+t=s\\}$. The same statement holds under the mass constraints $\\int_{Q_+}w=M$ and $\\int_{Q_+}u=m$, and then the minima of the constrained phase-field problems converge to the sharp constrained minimum.","pith_inferences":["An unconditional compactness result would need a coupling between $u_\\varepsilon$ and the interfacial layer, such as an $L^2$ or $L^1$ penalization of $u_\\varepsilon$ away from the transition region; without it, bounded energy allows the adatom mass to concentrate off the interface, as the example $\\psi\\equiv1$, $u_\\varepsilon=m/\\varepsilon$ on the layer shows.","The same splitting-at-a-cut argument could be transplanted to higher-dimensional epitaxial models once one has a local $\\Gamma$-convergence statement for pairs of measures and energy measures, suggesting the result is not tied to the one-dimensional graph setting.","Numerically, the theorem justifies using the smooth constrained functional as a computational proxy: choose $\\varepsilon$ small compared with island features, enforce the mass constraints on $w$ and $u$, and minimize; the $\\Gamma$-limit guarantees that sharp equilibria are recovered in the limit.","The infimal convolution $\\psi^c$ on cuts could be read as a selection rule for fracture patterns: when adatom densities are high, the energetically cheapest way to open a crack is to split the density between the two sides, a tendency that may appear in phase-field simulations as symmetric crack openings."],"forward_implications":["For every admissible sharp configuration $(\\Omega,v,\\mu)$ with finite energy and fixed mass $m,M$, there is a recovery sequence of phase-field triples respecting the exact mass constraints whose energy converges to the sharp value, so the constrained phase-field problem is a faithful variational approximation.","Any sequence of minimizers of $F_\\varepsilon^{m,M}$ with bounded elastic energy and with adatom measures converging to an interface-supported measure clusters at minimizers of $F^{m,M}$, and the minimal energies converge.","The phase-field limit sees fractures: the cut part of a BV graph contributes $\\psi^c(u)$ per unit length, so vertical cuts inside the film are energetically visible even though the approximating sequence uses smooth functions.","The approximation works for any Borel cost $\\psi\\colon[0,\\infty)\\to(0,\\infty)$ with positive infimum; the convexity and sub-additivity that the sharp limit needs are generated by the $\\Gamma$-limit itself, not imposed on the approximating densities.","The role of $\\eta_\\varepsilon=o(\\varepsilon)$ is to provide compactness for the displacement and phase-field sequences; removing it would jeopardize the convergence of minimizing sequences."],"supporting_citations":[{"why":"It supplies the sharp relaxed functional $F$ and the relaxation and approximation results, including the wriggling process, that the phase-field limit must reproduce.","marker":"[9]"},{"why":"It provides the general $\\Gamma$-convergence theorem for surface energies with adatom densities that is used as the local liminf tool in Theorem 9.","marker":"[5]"},{"why":"It supplies the phase-field construction for epitaxially strained films and the strategy for matching the bulk and surface terms in the recovery sequence.","marker":"[3]"},{"why":"It provides the almost optimal profile problem and the Modica–Mortola limsup estimate used to build the phase-field profile.","marker":"[20]"},{"why":"It introduced the adatom-density modelling and the wriggling process that the later relaxation and the present proof rely on.","marker":"[6]"}],"fun_headline_variants":["Adatom phase-field converges to sharp epitaxial model","Mass-constrained phase-field limits to sharp film energy","Adatom density encoded, sharp limit recovered for epitaxy","Phase-field with adatoms Γ-limits to sharp relaxed energy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes—and its own Remark 6 says the assumption is essential—that the adatom measures from bounded-energy phase-field sequences have weak-* limits supported on the limiting film interface; this support condition is what lets the mass constraint pass to the limit and is not a consequence of the energy bounds.","fun_headline_variants_meta":{"raw":{"variants":["Adatom phase-field converges to sharp epitaxial model","Mass-constrained phase-field limits to sharp film energy","Adatom density encoded, sharp limit recovered for epitaxy","Phase-field with adatoms Γ-limits to sharp relaxed energy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000582,"raw_usage":{"total_tokens":2732,"prompt_tokens":933,"completion_tokens":1799,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":1732}},"tokens_in":549,"tokens_out":1799,"duration_ms":11949,"temperature":1.0,"reasoning_tokens":1732,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:09:55.007883+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\psi\\equiv1$ and put $u_\\varepsilon=m/\\varepsilon$ on the $\\varepsilon$-transition layer of $w_\\varepsilon$; the surface-energy term of $F_\\varepsilon$ stays bounded while $\\mu_\\varepsilon$ has mass of order $1/\\varepsilon$, so no subsequence can weak-* converge with support on the interface. This calculation makes concrete the failure flagged in Remark 6: without hypothesis (12), the compactness asserted in the equi-coercivity step behind Theorem 7 cannot hold.","supporting_citations":[{"cited_title":"Cristoferi and G","cited_arxiv_id":null,"evidence_quote":"It supplies the sharp relaxed functional $F$ and the relaxation and approximation results, including the wriggling process, that the phase-field limit must reproduce."},{"cited_title":"Caroccia and R","cited_arxiv_id":null,"evidence_quote":"It provides the general $\\Gamma$-convergence theorem for surface energies with adatom densities that is used as the local liminf tool in Theorem 9."},{"cited_title":"Bonnetier and A","cited_arxiv_id":null,"evidence_quote":"It supplies the phase-field construction for epitaxially strained films and the strategy for matching the bulk and surface terms in the recovery sequence."},{"cited_title":"Modica , The gradient theory of phase transitions and the minimal interface criterion , Arch","cited_arxiv_id":null,"evidence_quote":"It provides the almost optimal profile problem and the Modica–Mortola limsup estimate used to build the phase-field profile."},{"cited_title":"Caroccia, R","cited_arxiv_id":null,"evidence_quote":"It introduced the adatom-density modelling and the wriggling process that the later relaxation and the present proof rely on."}],"review_version":1}