Minimizing movements for anisotropic and inhomogeneous mean curvature flows

References

[1] S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing, Acta Metall. 27 (1979), no. 5, 1085–1095. 10.1016/0001-6160(79)90196-2Search in Google Scholar

[2] L. Almeida, A. Chambolle and M. Novaga, Mean curvature flow with obstacles, Ann. Inst. H. Poincaré C Anal. Non Linéaire 29 (2012), no. 5, 667–681. 10.1016/j.anihpc.2012.03.002Search in Google Scholar

[3] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim. 31 (1993), no. 2, 387–438. 10.1137/0331020Search in Google Scholar

[4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, New York, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar

[5] G. Barles, An introduction to the theory of viscosity solutions for first-order Hamilton–Jacobi equations and applications, Hamilton–Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math. 2074, Springer, Heidelberg (2013), 49–109. 10.1007/978-3-642-36433-4_2Search in Google Scholar

[6] G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), no. 2, 439–469. 10.1137/0331021Search in Google Scholar

[7] G. Barles and P. E. Souganidis, A new approach to front propagation problems: Theory and applications, Arch. Ration. Mech. Anal. 141 (1998), no. 3, 237–296. 10.1007/s002050050077Search in Google Scholar

[8] G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J. 25 (1996), no. 3, 537–566. 10.14492/hokmj/1351516749Search in Google Scholar

[9] F. Cagnetti, M. G. Mora and M. Morini, A second order minimality condition for the Mumford–Shah functional, Calc. Var. Partial Differential Equations 33 (2008), no. 1, 37–74. 10.1007/s00526-007-0152-3Search in Google Scholar

[10] A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound. 6 (2004), no. 2, 195–218. 10.4171/ifb/97Search in Google Scholar

[11] A. Chambolle, M. Morini, M. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc. 32 (2019), no. 3, 779–824. 10.1090/jams/919Search in Google Scholar

[12] A. Chambolle, M. Morini and M. Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal. 44 (2012), no. 6, 4048–4077. 10.1137/120863587Search in Google Scholar

[13] A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal. 218 (2015), no. 3, 1263–1329. 10.1007/s00205-015-0880-zSearch in Google Scholar

[14] A. Chambolle and M. Novaga, Implicit time discretization of the mean curvature flow with a discontinuous forcing term, Interfaces Free Bound. 10 (2008), no. 3, 283–300. 10.4171/ifb/190Search in Google Scholar

[15] A. Chambolle and M. Novaga, Anisotropic and crystalline mean curvature flow of mean-convex sets, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 2, 623–643. 10.2422/2036-2145.202005_009Search in Google Scholar

[16] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. 10.4310/jdg/1214446564Search in Google Scholar

[17] G. De Philippis and T. Laux, Implicit time discretization for the mean curvature flow of mean convex sets, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 21 (2020), 911–930. 10.2422/2036-2145.201810_003Search in Google Scholar

[18] G. De Philippis and F. Maggi, Regularity of free boundaries in anisotropic capillarity problems and the validity of Young’s law, Arch. Ration. Mech. Anal. 216 (2015), no. 2, 473–568. 10.1007/s00205-014-0813-2Search in Google Scholar

[19] M. Desbrun, M. Meyer, P. Schröder and A. H. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow, Proceedings of the 26th annual conference on Computer graphics and interactive techniques, ACM, New York (1999), 317–324. 10.1145/311535.311576Search in Google Scholar

[20] S. Esedoḡlu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions, Comm. Pure Appl. Math. 68 (2015), no. 5, 808–864. 10.1002/cpa.21527Search in Google Scholar

[21] T. Eto, Y. Giga and K. Ishii, An area-minimizing scheme for anisotropic mean-curvature flow, Adv. Differential Equations 17 (2012), no. 11–12, 1031–1084. 10.57262/ade/1355702938Search in Google Scholar

[22] L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), no. 9, 1097–1123. 10.1002/cpa.3160450903Search in Google Scholar

[23] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. 10.4310/jdg/1214446559Search in Google Scholar

[24] J. Fuchs and T. Laux, Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets, preprint (2022), https://arxiv.org/abs/2201.00413. Search in Google Scholar

[25] N. Fusco, V. Julin and M. Morini, Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, J. Geom. Anal. 32 (2022), no. 2, Paper No. 53. 10.1007/s12220-021-00806-xSearch in Google Scholar

[26] Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 (1991), no. 2, 443–470. 10.1512/iumj.1991.40.40023Search in Google Scholar

[27] Y. Giga and N. Požár, Motion by crystalline-like mean curvature: a survey, Bull. Math. Sci. 12 (2022), no. 2, Paper No. 2230004. 10.1142/S1664360722300043Search in Google Scholar

[28] M. E. Gurtin, Toward a nonequilibrium thermodynamics of two-phase materials, Arch. Ration. Mech. Anal. 100 (1988), no. 3, 275–312. 10.1007/BF00251518Search in Google Scholar

[29] S. Hensel and T. Laux, A new varifold solution concept for mean curvature flow: Convergence of the Allen–Cahn equation and weak-strong uniqueness, preprint (2021), https://arxiv.org/abs/2109.04233. Search in Google Scholar

[30] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463–480. 10.1007/BF01388742Search in Google Scholar

[31] G. Huisken and A. Polden, Geometric evolution equations for hypersurfaces, Calculus of Variations and Geometric Evolution Problems (Cetraro 1996), Lecture Notes in Math. 1713, Springer, Berlin (1999), 45–84. 10.1007/BFb0092669Search in Google Scholar

[32] J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J. 35 (1986), no. 1, 45–71. 10.1512/iumj.1986.35.35003Search in Google Scholar

[33] T. Ilmanen, Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), no. 2, 417–461. 10.4310/jdg/1214454300Search in Google Scholar

[34] T. Ilmanen, The level-set flow on a manifold, Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles 1990), Proc. Sympos. Pure Math. 54, American Mathematical Society, Providence (1993), 193–204. 10.1090/pspum/054.1/1216585Search in Google Scholar

[35] H. Ishii and P. Souganidis, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku Math. J. (2) 47 (1995), no. 2, 227–250. 10.2748/tmj/1178225593Search in Google Scholar

[36] T. Laux and F. Otto, Convergence of the thresholding scheme for multi-phase mean-curvature flow, Calc. Var. Partial Differential Equations 55 (2016), no. 5, Article ID 129. 10.1007/s00526-016-1053-0Search in Google Scholar

[37] T. Laux, K. Stinson and C. Ullrich, Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow, preprint (2022), https://arxiv.org/abs/2212.11939. Search in Google Scholar

[38] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 253–271. 10.1007/BF01205007Search in Google Scholar

[39] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Stud. Adv. Math. 135, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139108133Search in Google Scholar

[40] B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple junctions: A level set approach, J. Comput. Phys. 112 (1994), no. 2, 334–363. 10.1006/jcph.1994.1105Search in Google Scholar

[41] L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 (2016), no. 1, Article ID 18. 10.1007/s00526-015-0943-xSearch in Google Scholar

[42] R. Schoen, L. Simon and F. J. Almgren, Jr., Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II, Acta Math. 139 (1977), no. 3–4, 217–265. 10.1007/BF02392238Search in Google Scholar

[43] J. A. Sethian, Level Set Methods and Fast Marching Methods, 2nd ed., Cambridge Monogr. Appl. Comput. Math. 3, Cambridge University, Cambridge, 1999. Search in Google Scholar

[44] G. Taubin, A signal processing approach to fair surface design, Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, ACM, New York (1995), 351–358. 10.1145/218380.218473Search in Google Scholar