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Equivalence of weak solution concepts for mean curvature flow
Open accessAbstract
We provide a connection between weak solution concepts of mean curvature flow. On the one side we have the viscosity solution which is based on the comparison principle. On the other, variational solutions, which include Ilmanen’s boundary motion as well as combined Brakke flows and distributional solutions. We prove that if one has a “foliation” by variational solutions, then the resulting function is the unique viscosity solution. This answers an open question suggested by the work of Evans and Spruck [J. Geom. Anal. 5 (1995), pp. 77–114], Ilmanen [Mem. Amer. Math. Soc. 108 (1994), pp. x+90], and the authors [J. Geom. Anal. 34 (2024), p. 18]. These results show that almost every level set of the viscosity solution is a variational solution. Thus, we establish the equivalence of these solution concepts. Moreover, we show the generic uniqueness of variational solutions.
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References 17 · Cited by 1
Research works printed but not linked: 19
- Alberti, Giovanni, 2013, Structure of level sets and Sard-type properties of Lipschitz maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12, 863
- Sigurd B. Angenent, Tom Ilmanen, and Juan J. L. Velázquez, Fattening from smooth initial data in mean curvature flow, Preprint, 2002.
- Brakke, Kenneth A., 1978, The motion of a surface by its mean curvature, 20
- Chambolle, Antonin, 2021, Anisotropic tubular neighborhoods of sets, Math. Z., 299, 1257, 10.1007/s00209-021-02715-9 · DOI
- Yun-Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differ. Geom. 33(3):749–786, 1991. DOI 10.4310/jdg/1214446564. · DOI
- Lawrence C. Evans and Joel Spruck, Motion of level sets by mean curvature. I, J. Differ. Geom. 33(3):635–681, 1991. DOI 10.4310/jdg/1214446559. · DOI
- Evans, Lawrence C., 1995, Motion of level sets by mean curvature. IV, J. Geom. Anal., 5, 77, 10.1007/BF02926443 · DOI
- Julian Fischer, Sebastian Hensel, Tim Laux, and Theresa M. Simon, The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions J. Eur. Math. Soc. 2025. DOI 10.4171/JEMS/1577. · DOI
- Hensel, Sebastian, 2025, A new varifold solution concept for mean curvature flow: convergence of the Allen-Cahn equation and weak-strong uniqueness, J. Differential Geom., 130, 209, 10.4310/jdg/1747065796 · DOI · Paper page
- Ilmanen, Tom, 1994, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108, x+90, 10.1090/memo/0520 · DOI
- Ishii, Hitoshi, 1995, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., 38, 101, 10.24546/0100499855 · DOI
- Juutinen, Petri, 2001, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33, 699, 10.1137/S0036141000372179 · DOI
- Laux, Tim, [2020] \copyright2020, The thresholding scheme for mean curvature flow and de Giorgi’s ideas for minimizing movements, 63, 10.2969/aspm/08510063 · DOI
- Luckhaus, Stephan, 1995, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3, 253, 10.1007/BF01205007 · DOI
- Maggi, Francesco, 2012, Sets of finite perimeter and geometric variational problems, 135, 10.1017/CBO9781139108133 · DOI
- Mugnai, Luca, 2011, Convergence of perturbed Allen-Cahn equations to forced mean curvature flow, Indiana Univ. Math. J., 60, 41, 10.1512/iumj.2011.60.3949 · DOI
- Stuvard, Salvatore, 2024, On the existence of canonical multi-phase Brakke flows, Adv. Calc. Var., 17, 33, 10.1515/acv-2021-0093 · DOI
- Ullrich, Anton, 2024, Generic level sets in mean curvature flow are BV solutions, J. Geom. Anal., 34, Paper No. 375, 18, 10.1007/s12220-024-01819-y · DOI
- White, Brian, 2024, The avoidance principle for noncompact hypersurfaces moving by mean curvature flow, Calc. Var. Partial Differential Equations, 63, Paper No. 111, 20, 10.1007/s00526-024-02725-5 · DOI
Metadata
- Authors
- Laux, T., & Ullrich, A.
- Venue
- Proceedings of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2026-08-17
- Volume
- 154
- Issue
- 12
- Pages
- 5347-5360
- ISSN
- 0002-9939 (ISSN-L)
- 1088-6826
- OpenAlex
- W4415953806
- Cited by
- 1
- References
- 17
- Citations by year
- 2026: 1
- FWCI
- 2.20
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Proceedings of the American Mathematical Society (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Applied Mathematics › Nonlinear Partial Differential Equations
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Keywords
- mean curvature flow, viscosity solutions, variational solutions, Brakke flow, distributional solutions, comparison principle, uniqueness of solutions, level sets, foliations
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