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Variational Nonlinear and Nonlocal Curvature Flows

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Pith's one-line read Nonlinear and nonlocal curvature flows with time-dependent forcing admit level-set viscosity solutions obtained as limits of a variational time-discrete scheme, and these solutions are unique when the curvature obeys a first-order or…

desk verdict Strong existence result for nonlinear nonlocal curvature flows, but the uniqueness proof has load-bearing sign errors that look repairable. read the letter →

arxiv 2506.05951 v1 pith:XNCGRPU3 submitted 2025-06-06 math.AP

Daniele De Gennaro This is my paper
classification math.AP MSC 35D4035K5535R1149Q20
keywords nonlocalcurvatureflowvariationalminimizingmovementslevel-setviscositysolutionscomparisonprinciplegeneralizedperimeterAlmgren–Taylor–Wangscheme
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes existence and uniqueness for a broad class of geometric evolutions in which a set moves by a nonlinear function of a nonlocal variational curvature, with time-dependent forcing and an anisotropic mobility. Existence is obtained by an Almgren–Taylor–Wang minimizing-movements scheme: the discrete evolutions converge, as the time step tends to zero, to a continuous viscosity solution of the level-set equation $u_t - \psi(\nabla u)\,G(-\kappa(x,\{u\ge u\})+f(t))=0$. The nonlinearity $G$ is only required to be continuous, non-decreasing, and vanishing at zero, so the result covers truncations, powers, and other modifications of classical mean curvature flow in a unified way. Uniqueness is proved under additional structural assumptions on the curvature alone, not on $G$. The paper thereby extends the linear variational-curvature theory to nonlinear speeds while keeping the variational approximation.

What carries the argument

The engine of the existence proof is the modified Almgren–Taylor–Wang functional $$F^E_{h,t}(F)=J(F)+\int_{E\triangle F}\left|g\left(\tfrac{\mathrm{sd}^\psi_E}{h}\right)\right|dx - f([t/h]h)\,|F|,$$ where $J$ is a submodular translation-invariant generalized perimeter, $\mathrm{sd}^\psi_E$ is the signed anisotropic distance from the set $E$, and $g$ is a selection of the (possibly multivalued) inverse of $G$. Iterating the minimal minimizers of this functional produces the discrete evolutions $u_h$. The convergence proof rests on two quantitative statements about balls: an upper bound on how fast a ball can grow (Lemma 2.13) and a lower bound on how fast it can shrink (Lemma 2.14), which together give the equicontinuity and compactness needed for Ascoli–Arzelà. Uniqueness is carried by the viscosity-solution comparison principle, proved by doubling variables and using either (FO) or (C') to compare the relaxed curvatures of the superlevel sets.

What would settle it

Take a specific submodular translation-invariant generalized perimeter, for instance a nonlocal perimeter with a non-radially-symmetric kernel, and compute the inner and outer first variations (2.10) at a point on a sphere; if the two limits differ, the variational curvature used by the paper is not defined and the main theorem's hypotheses are not met.

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Extended reading notes

Core claim

The central claim is that the discrete-in-time minimizing-movements functions $u_h$, built by iterating the scheme, converge uniformly on compact time sets to a continuous viscosity solution of the Cauchy problem $$u_t - \psi(\nabla u)\,G\!\left(-\kappa(x,\{u\ge u\}) + f(t)\right)=0,$$ where $\kappa$ is a variational curvature, $\psi$ is a mobility, $f$ is a bounded continuous forcing, and $G$ is any continuous non-decreasing function with $G(0)=0$. This is the content of Theorem 3.5. Under the additional assumption that the curvature is of first order (condition (FO)) or satisfies a strengthened uniform-regularity condition (C'), the paper proves a comparison principle for sub- and supersolutions (Theorem 4.9), which implies that the level-set evolution is unique and independent of the approximating sequence.

Load-bearing premise

The argument assumes without proof that every $C^2$ set carries a well-defined variational curvature: the inner and outer first variations in Definition 2.7 agree and produce a continuous, translation-invariant function satisfying the lower ball bound (D), and the later estimates and comparison principle all rest on that assumption.

Editorial extensions

If this is right

  • For the nonlinearities named in the introduction—truncated speeds $G(s)=(-M)\vee s\wedge M$, the affine-invariant power $G(s)=s^{1/3}$, and purely shrinking evolutions—the theorem supplies a well-posed level-set flow together with a convergent variational approximation.
  • The time-discrete scheme is constructive: each step solves the incremental problem (2.11), so it can be used as a numerical algorithm for nonlinear and nonlocal evolutions.
  • When the curvature satisfies (FO) or (C'), the comparison principle makes the level-set solution unique; in particular the limit does not depend on the choice of subsequence or on the particular selection of minimizers.
  • The forcing term $f$ may vary continuously in time without extra regularity, which widens the range of applications compared with the constant-forcing case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same scheme could plausibly be adapted to forcing that depends on the spatial variable, although the uniqueness proof would need a new ingredient because the spatial constancy of $f$ is used in the doubling-of-variables step (4.19).
  • The ball-speed estimates (Lemmas 2.13–2.14) might yield explicit quantitative bounds on the discrete speed for fractional perimeters, which could be useful for numerical simulation of fractional mean curvature flow.
  • Because $G$ is not required to be Lipschitz, the framework may allow discontinuous nonlinearities after a relaxation step; whether the minimal assumptions (continuity, monotonicity, $G(0)=0$) are necessary is a natural testable question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper studies the nonlinear nonlocal curvature flow (1.1) with variational curvatures, time-dependent forcing, and mobility. The author proves (Theorem 3.5) that the Almgren-Taylor-Wang minimizing movements scheme converges, up to subsequences, to a continuous viscosity solution of the level-set equation (2.5), assuming only continuity, monotonicity, and G(0)=0. Under either the first-order condition (FO) or the strengthened continuity condition (C'), the author claims a comparison principle (Theorem 4.9) yielding uniqueness of level-set solutions. The existence argument follows the architecture of [16] and [11] with an approximation procedure to handle bounded nonlinearities. The uniqueness section adapts the viscosity techniques of [16] to the nonlinear and time-dependent setting.

Significance. Should the results hold, the paper would extend the variational approach to curvature flows to a broad class of nonlinear evolution laws with minimal regularity assumptions on the nonlinearity, covering truncated MCF, powers of mean curvature, and nonlocal curvatures such as fractional perimeter. The convergence theorem is a substantial piece of work, and the paper is transparent in stating its standing assumptions and in citing the unpublished preprint [10] as its starting point. The uniqueness result is of independent interest and would generalize the comparison principles of [16] to nonlinear G and time-dependent forcing. However, the printed proof of the uniqueness theorem contains sign inconsistencies that currently prevent Theorem 4.9 from being established.

major comments (2)
  1. [4.2 (Theorem 4.9, FO case), Eq. (4.5)] In the (FO) case, the second inequality in (4.5) is written with '≤ 0', but the jet listed there is in P^{2,-}v(yβ,sβ), so Lemma 4.5 for supersolutions yields '≥ 0'. With both inequalities as printed, inequality (4.6) does not follow, and the contradiction argument in this case collapses.
  2. [4.2 (Theorem 4.9, C' case), Eqs. (4.15) and (4.17)] The inequalities (4.15) and (4.17) are written in a form opposite to Lemma 4.5 and to the level-set equation (2.5). For the subsolution one needs ˘a - ψ(|p|)G(-κ* + f) ≤ 0, and for the supersolution ˘a - 2ε - ψ(|p|)G(-κ_* + f) ≥ 0, whereas the paper has +ψ(|p|)G(κ* + f) in both places. Consequently (4.18) cannot be derived, and the (C') case is unproven as written. These are likely repairable sign errors, but they are load-bearing for the uniqueness claim.
minor comments (4)
  1. [4.2, before Eq. (4.5)] In (4.5) and the preceding line, 'αf'(|pβ|)' should read 'αℓ'(|pβ|)', since f is the time-dependent forcing and ℓ is the auxiliary function appearing in the penalty αℓ(|x-y|).
  2. [Definition 2.4] In the admissibility condition, the term 'ηt(z)(t - t)' should be 'ηt(z)(t - \hat t)'.
  3. [3.21] In the estimate for the extinction time, 'g(t) → 0' should presumably be 'β(t) → 0'.
  4. [References] The reference [11] is cited without a title; please provide the full bibliographic data.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the minimizing-movements convergence and the comparison principle are derived from explicit axioms and published prior results, not from their conclusions.

full rationale

The derivation chain is self-contained with respect to the paper's own claims. The minimizing-movements convergence (Theorem 3.5) is proved from the variational definition of kappa (Definition 2.7), the axioms (A)-(D), and the stated hypotheses on G and f; the discrete-to-continuous passage uses ball estimates (Lemmas 2.13-2.14), equicontinuity, and Ascoli-Arzelà, with no fitted parameter or quantity renamed as a prediction. The uniqueness theorem (Theorem 4.9) is presented as an adaptation of comparison principles from [16] under the explicitly stated additional hypotheses (FO) or (C'), and Lemma 4.5 is derived from the viscosity definitions rather than assumed. The paper cites the author's own prior work [11] for technical estimates in the convergence proof, but those are published external estimates and do not include the target theorem, so the self-citation is not load-bearing in a circular sense. The unpublished preprint [10] is openly identified only as a starting point, not as a substitute for the proof. The alleged sign inconsistencies in the printed comparison proof are correctness concerns rather than circularity: they do not make the conclusion equivalent to an input of the argument.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical or empirical parameters are fitted, so the free-parameter count is zero. The constants c_psi, K, a and b are structural bounds, not fitted values. The axioms are the defining framework of variational curvatures plus the explicit regularity assumptions for uniqueness.

assumptions (6)
  • domain assumption The generalized perimeter J satisfies submodularity, translation invariance, lower semicontinuity, and finiteness on C^2 sets (Definition 2.6).
    This is the variational backbone of the minimizing movements scheme; introduced in [16], Section 2.1.
  • domain assumption A variational curvature kappa(x,E) exists as the common value of the one-sided first variations kappa+ and kappa- for every C^2 set, and satisfies continuity (C) and the ball lower bound (D).
    Assumed in Section 2.1 after Definition 2.7; controls the discrete speed estimates in Lemmas 2.13 and 2.14.
  • domain assumption G is continuous, non-decreasing, G(0)=0, with extended limits -a and b; f is bounded and continuous.
    Definition of the flow in (2.4); these minimal regularity assumptions are the paper's target level of generality.
  • domain assumption The anisotropy psi is convex, even, positively 1-homogeneous and comparable to the Euclidean norm.
    Definition 2.1; ensures the signed psi-distance is Lipschitz equivalent to Euclidean distance.
  • domain assumption For uniqueness, kappa satisfies either the first-order condition (FO) or the strengthened uniform regularity condition (C').
    Conditions stated in Section 4.2; the comparison principle Theorem 4.9 is proved under one of these.
  • standard math Standard viscosity solution machinery: admissible test functions with the family L, parabolic semijets, Jensen's lemma, and doubling of variables.
    Used throughout Sections 3.1 and 4.2, following Giga and Crandall-Ishii-Lions.

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Cite this review

Pith. "Pith review of Variational Nonlinear and Nonlocal Curvature Flows." pith.science (2026). https://pith.science/paper/XNCGRPU3

@misc{pith2026250605951,
  author       = {Pith},
  title        = {Pith review of: Variational Nonlinear and Nonlocal Curvature Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XNCGRPU3}},
  note         = {Machine review of arXiv:2506.05951}
}
read the original abstract

We prove that the minimizing movements scheme \'a la Almgren-Taylor-Wang converges towards level-set solutions to a nonlinear version of nonlocal curvature flows with time-depending forcing term, in the rather general framework of variational curvatures introduced in \cite{ChaMorPon15}. The nonlinearity involved is assumed to satisfy minimal assumptions, namely continuity, monotonicity, and vanishing at zero. Under additional assumptions only on the curvatures involved, we establish uniqueness for level-set solutions.

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Works this paper leans on

33 extracted references · 23 canonical work pages

  1. [16]

    Nonlocal curvature flows

    A. Chambolle, M. Morini, and M. Ponsiglione. “Nonlocal curvature flows”. In: Arch. Ration. Mech. Anal. 218.3 (2015), pp. 1263–1329. doi: 10.1007/s00205-015-0880-z

    doi:10.1007/s00205-015-0880-z 2015

  2. [11]

    Chambolle, D

    A. Chambolle, D. De Gennaro, and M. Morini. In: Advances in Calculus of Variations (2023). doi: doi:10.1515/acv-2022-0102. url: https://doi.org/10.1515/acv-2022-0102

    work page doi:10.1515/acv-2022-0102 2023

  3. [10]

    Nonlinear Mean Curvature Flow

    A. Chambolle, A. Ciomaga, and G. Thoroude. “Nonlinear Mean Curvature Flow”. Unpub- lished work

    work page

  4. [1]

    Curvature-driven flows: a variational approach

    F. Almgren, J. E. Taylor, and L. Wang. “Curvature-driven flows: a variational approach”. In: SIAM J. Control Optim. 31.2 (1993), pp. 387–438. issn: 0363-0129. doi: 10.1137/0331020

    doi:10.1137/0331020 1993

  5. [2]

    Axioms and fundamental equations of image processing

    L. Alvarez, F. Guichard, P.-L. Lions, and J.-M. Morel. “Axioms and fundamental equations of image processing”. In: Arch. Rational Mech. Anal. 123.3 (1993), pp. 199–257. issn: 0003-9527. doi: 10.1007/BF00375127. url: https://doi.org/10.1007/BF00375127

    work page doi:10.1007/bf00375127 1993

  6. [3]

    Moving surfaces by non-concave curvature functions

    B. Andrews. “Moving surfaces by non-concave curvature functions”. In: Calc. Var. Partial Differential Equations 39.3-4 (2010), pp. 649–657. issn: 0944-2669. doi: 10.1007/s00526- 010-0329-z

    doi:10.1007/s00526- 2010

  7. [4]

    Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

    D. Azagra, M. Jim´ enez-Sevilla, and F. Maci` a. “Generalized motion of level sets by functions of their curvatures on Riemannian manifolds”. In: Calc. Var. Partial Differential Equations 33.2 (2008), pp. 133–167. issn: 0944-2669. doi: 10.1007/s00526-008-0160-y . url: https: //doi.org/10.1007/s00526-008-0160-y

    work page doi:10.1007/s00526-008-0160-y 2008

  8. [5]

    Bellettini and S

    G. Bellettini and S. Kholmatov. Minimizing movements for the generalized power mean cur- vature flow. cvgmt preprint. 2024. url: http://cvgmt.sns.it/paper/6505/

    work page 2024

Show all 33 references
  1. [6]

    Front propagation problems with nonlocal terms. II

    Pierre Cardaliaguet. “Front propagation problems with nonlocal terms. II”. In: J. Math. Anal. Appl. 260.2 (2001), pp. 572–601. issn: 0022-247X. doi: 10. 1006/ jmaa. 2001. 7483. url: https://doi.org/10.1006/jmaa.2001.7483. REFERENCES 29

    2001

  2. [7]

    On front propagation problems with nonlocal terms

    Pierre Cardaliaguet. “On front propagation problems with nonlocal terms”. In: Adv. Differ- ential Equations 5.1-3 (2000), pp. 213–268. issn: 1079-9389

    2000

  3. [8]

    Some flows in shape optimization

    Pierre Cardaliaguet and Olivier Ley. “Some flows in shape optimization”. In: Arch. Ration. Mech. Anal. 183.1 (2007), pp. 21–58. issn: 0003-9527. doi: 10.1007/s00205- 006- 0002- z. url: https://doi.org/10.1007/s00205-006-0002-z

    2007 doi

  4. [9]

    An algorithm for mean curvature motion

    A. Chambolle. “An algorithm for mean curvature motion”. In: Interfaces Free Bound. 6.2 (2004), pp. 195–218. issn: 1463-9963. doi: 10.4171/IFB/97 . url: https://doi.org/10. 4171/IFB/97

    2004 doi

  5. [12]

    Continuous limits of discrete perimeters

    A. Chambolle, A. Giacomini, and L. Lussardi. “Continuous limits of discrete perimeters”. In: M2AN Math. Model. Numer. Anal. 44.2 (2010), pp. 207–230. issn: 0764-583X. doi: 10.1051/ m2an/2009044. url: https://doi.org/10.1051/m2an/2009044

    2010

  6. [13]

    Existence and uniqueness for anisotropic and crystalline mean curvature flows

    A. Chambolle, M. Morini, M. Novaga, and M. Ponsiglione. “Existence and uniqueness for anisotropic and crystalline mean curvature flows”. In: J. Amer. Math. Soc. 32.3 (2019), pp. 779–824. issn: 0894-0347. doi: 10.1090/jams/919 . url: https://doi.org/10.1090/ jams/919

    2019 doi

  7. [14]

    Generalized crystalline evolutions as limits of flows with smooth anisotropies

    A. Chambolle, M. Morini, M. Novaga, and M. Ponsiglione. “Generalized crystalline evolutions as limits of flows with smooth anisotropies”. In: Anal. PDE 12.3 (2019), pp. 789–813. issn: 2157-5045. doi: 10.2140/apde.2019.12.789

    2019 doi

  8. [15]

    Existence and uniqueness for a crystalline mean curvature flow

    A. Chambolle, M. Morini, and M. Ponsiglione. “Existence and uniqueness for a crystalline mean curvature flow”. In: Comm. Pure Appl. Math. 70.6 (2017), pp. 1084–1114. issn: 0010-

    2017

  9. [17]

    Implicit time discretization of the mean curvature flow with a discontinuous forcing term

    A. Chambolle and M. Novaga. “Implicit time discretization of the mean curvature flow with a discontinuous forcing term”. In: Interfaces Free Bound. 10.3 (2008), pp. 283–300. issn: 1463-9963. doi: 10.4171/ifb/190. url: https://doi.org/10.4171/ifb/190

    2008 doi

  10. [18]

    Deforming convex hypersurfaces by the square root of the scalar curvature

    B. Chow. “Deforming convex hypersurfaces by the square root of the scalar curvature”. In: Invent. Math. 87.1 (1987), pp. 63–82. issn: 0020-9910. doi: 10 . 1007 / BF01389153. url: https://doi.org/10.1007/BF01389153

    1987 doi

  11. [19]

    User’s guide to viscosity solutions of second order partial differential equations

    Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. “User’s guide to viscosity solutions of second order partial differential equations”. In: Bull. Amer. Math. Soc. (N.S.) 27.1 (1992), pp. 1–67. issn: 0273-0979. doi: 10.1090/S0273-0979-1992-00266-5 . url: https://doi. ...

    1992 doi

  12. [20]

    Time-fractional Allen–Cahn equations versus powers of the mean curvature

    S. Dipierro, M. Novaga, and E. Valdinoci. “Time-fractional Allen–Cahn equations versus powers of the mean curvature”. In: Physica D: Nonlinear Phenomena 463 (2024), p. 134172

    2024

  13. [21]

    Y. Giga. Surface evolution equations . Vol. 99. Monographs in Mathematics. A level set ap- proach. Birkh¨ auser Verlag, Basel, 2006, pp. xii+264

    2006

  14. [22]

    A level set crystalline mean curvature flow of surfaces

    Y. Giga and N. Poˇ z´ ar. “A level set crystalline mean curvature flow of surfaces”. In: Adv. Differential Equations 21.7-8 (2016), pp. 631–698. issn: 1079-9389

    2016

  15. [23]

    Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow

    Y. Giga and N. Poˇ z´ ar. “Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow”. 30 REFERENCES In: Comm. Pure Appl. Math. 71.7 (2018), pp. 1461–1491. issn: 0010-3640. doi:...

    2018

  16. [24]

    Level set approach for fractional mean curvature flows

    C. Imbert. “Level set approach for fractional mean curvature flows”. In: Interfaces Free Bound. 11.1 (2009), pp. 153–176. issn: 1463-9963. doi: 10.4171/IFB/207

    2009 doi

  17. [25]

    Implicit time discretization for the mean curvature flow equation

    S. Luckhaus and T. Sturzenhecker. “Implicit time discretization for the mean curvature flow equation”. In: Calc. Var. Partial Differential Equations 3.2 (1995), pp. 253–271. issn: 0944-

    1995

  18. [26]

    Level set and variational methods for geometric flows

    M. Morini. “Level set and variational methods for geometric flows”. In: 2022-2023 MATRIX Annals. MATRIX Book Series (to appear)

    2022

  19. [27]

    Non-homothetic convex ancient solutions for flows by high powers of curvature

    Susanna Risa and Carlo Sinestrari. “Non-homothetic convex ancient solutions for flows by high powers of curvature”. In: Ann. Mat. Pura Appl. (4) 202.2 (2023), pp. 601–618. issn: 0373-3114. doi: 10.1007/s10231-022-01253-3 . url: https://doi.org/10.1007/s10231- 022-01253-3

    2023 doi

  20. [28]

    Affine invariant scale-space

    G. Sapiro and A. Tannenbaum. “Affine invariant scale-space”. In: International journal of computer vision 11.1 (1993), pp. 25–44

    1993

  21. [29]

    Evolution of convex hypersurfaces by powers of the mean curvature

    F. Schulze. “Evolution of convex hypersurfaces by powers of the mean curvature”. In: Math. Z. 251.4 (2005), pp. 721–733. issn: 0025-5874. doi: 10.1007/s00209-004-0721-5

    2005 doi

  22. [30]

    Nonlinear evolution by mean curvature and isoperimetric inequalities

    F. Schulze. “Nonlinear evolution by mean curvature and isoperimetric inequalities”. In: J. Differential Geom. 79.2 (2008), pp. 197–241. issn: 0022-040X

    2008

  23. [31]

    Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions

    Dejan Slepˇ cev. “Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions”. In: Nonlinear Anal. 52.1 (2003), pp. 79–115. issn: 0362- 546X. doi: 10.1016/S0362- 546X(02)00098- 6. url: https://doi.org/10.1016/S0362- 546X(02)00098-6...

    2003 doi

  24. [2669]

    doi: 10.1007/BF01205007

    doi

  25. [3640]

    doi: 10.1002/cpa.21668

    doi

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