The existence of a weak solution to volume preserving mean curvature flow in higher dimensions. (English) Zbl 07687726
Summary: In this paper, we construct a family of integral varifolds, which is a global weak solution to the volume preserving mean curvature flow in the sense of \(L^2\)-flow. This flow is also a distributional BV-solution for a short time, when the perimeter of the initial data is sufficiently close to that of a ball with the same volume. To construct the flow, we use the Allen-Cahn equation with a non-local term motivated by studies of Mugnai, Seis, and Spadaro, and Kim and Kwon. We prove the convergence of the solution for the Allen-Cahn equation to the family of integral varifolds with only natural assumptions for the initial data.
MSC:
| 35D30 | Weak solutions to PDEs |
| 35K93 | Quasilinear parabolic equations with mean curvature operator |
| 53E10 | Flows related to mean curvature |
Keywords:
family of integral varifolds; distributional BV-solution; Allen-Cahn equation with nonlocal termReferences:
| [1] | Almgren, F.; Taylor, JE; Wang, L., Curvature-driven flows: a variational approach, SIAM J. Control. Optim., 31, 387-438 (1993) · Zbl 0783.35002 · doi:10.1137/0331020 |
| [2] | Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (2000), New York: The Clarendon Press, Oxford University Press, New York · Zbl 0957.49001 |
| [3] | Antonopoulou, D.; Karali, G.; Sigal, IM, Stability of spheres under volume-preserving mean curvature flow, Dyn. Partial Differ. Equ., 7, 327-344 (2010) · Zbl 1219.35132 · doi:10.4310/DPDE.2010.v7.n4.a3 |
| [4] | Athanassenas, M., Volume-preserving mean curvature flow of rotationally symmetric surfaces, Comment. Math. Helv., 72, 52-66 (1997) · Zbl 0873.35033 · doi:10.1007/PL00000366 |
| [5] | Brakke, KA, The Motion of a Surface by Its Mean Curvature (1978), Princeton: Princeton University Press, Princeton · Zbl 0386.53047 |
| [6] | Brassel, M.; Bretin, E., A modified phase field approximation for mean curvature flow with conservation of the volume, Math. Methods Appl. Sci., 34, 1157-1180 (2011) · Zbl 1235.49082 · doi:10.1002/mma.1426 |
| [7] | Bronsard, L.; Stoth, B., Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28, 769-807 (1997) · Zbl 0874.35009 · doi:10.1137/S0036141094279279 |
| [8] | Chen, X.; Hilhorst, D.; Logak, E., Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound, 12, 527-549 (2010) · Zbl 1219.35018 · doi:10.4171/IFB/244 |
| [9] | Delfour, MC; Zolésio, JP, Shape analysis via oriented distance functions, J. Funct. Anal., 123, 129-201 (1994) · Zbl 0814.49032 · doi:10.1006/jfan.1994.1086 |
| [10] | Escher, J.; Simonett, G., The volume preserving mean curvature flow near spheres, Proc. Am. Math. Soc., 126, 2789-2796 (1998) · Zbl 0909.53043 · doi:10.1090/S0002-9939-98-04727-3 |
| [11] | Evans, LC; Gariepy, RF, Measure Theory and Fine Properties of Functions (2015), Boca Raton: CRC Press, Boca Raton · Zbl 1310.28001 · doi:10.1201/b18333 |
| [12] | Fischer, J.,Hensel, S.,Laux, T.,Simon, T.: The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions, arXiv preprint arXiv:2003.05478, 2020 |
| [13] | Fusco, N., The classical isoperimetric theorem, Rend. Accad. Sci. Fis. Mat. Napoli., 71, 63-107 (2004) · Zbl 1096.49024 |
| [14] | Fusco, N.; Maggi, F.; Pratelli, A., The sharp quantitative isoperimetric inequality, Ann. Math., 168, 941-980 (2008) · Zbl 1187.52009 · doi:10.4007/annals.2008.168.941 |
| [15] | Gage, M., On an area-preserving evolution equation for plane curves, Nonlinear problems in geometry (Mobile, Ala.,: Contemp. Math., 51 Amer. Math. Soc. Providence, R I1986, 51-62, 1985 · Zbl 0608.53002 |
| [16] | Giusti, E., Minimal Surfaces and Functions of Bounded Variation (1984), Boston: Birkhäuser, Boston · Zbl 0545.49018 · doi:10.1007/978-1-4684-9486-0 |
| [17] | Golovaty, D., The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Quart. Appl. Math., 55, 243-298 (1997) · Zbl 0878.35059 · doi:10.1090/qam/1447577 |
| [18] | Hensel, S., Laux, T.: A new varifold solution concept for mean curvature flow: convergence of the Allen-Cahn equation and weak-strong uniqueness, arXiv preprint arXiv:2109.04233, 2021 |
| [19] | Huisken, G., The volume preserving mean curvature flow, J. Reine Angew. Math., 382, 35-48 (1987) · Zbl 0621.53007 |
| [20] | Huisken, G., Asymptotic behavior for singularities of the mean curvature flow, J. Differ. Geom., 31, 285-299 (1990) · Zbl 0694.53005 · doi:10.4310/jdg/1214444099 |
| [21] | Hutchinson, JE, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J., 35, 45-71 (1986) · Zbl 0561.53008 · doi:10.1512/iumj.1986.35.35003 |
| [22] | Hutchinson, JE; Tonegawa, Y., Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory, Calc. Var. Partial Differ. Equ., 10, 49-84 (2000) · Zbl 1070.49026 · doi:10.1007/PL00013453 |
| [23] | Ilmanen, T., Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differ. Geom., 38, 417-461 (1993) · Zbl 0784.53035 · doi:10.4310/jdg/1214454300 |
| [24] | Kim, I.; Kwon, D., Volume preserving mean curvature flow for star-shaped sets, Calc. Var. Partial Differ. Equ. (2020) · Zbl 1445.35205 · doi:10.1007/s00526-020-01738-0 |
| [25] | Laux, T.: Weak-strong uniqueness for volume-preserving mean curvature flow, arXiv preprint arXiv:2205.13040, 2022 · Zbl 1404.65121 |
| [26] | Laux, T.; Otto, F., Convergence of the thresholding scheme for multi-phase mean-curvature flow, Calc. Var. Partial Differ. Equ. (2016) · Zbl 1388.35121 · doi:10.1007/s00526-016-1053-0 |
| [27] | Laux, T.; Simon, TM, Convergence of the Allen-Cahn equation to multiphase mean curvature flow, Comm. Pure Appl. Math., 71, 1597-1647 (2018) · Zbl 1393.35122 · doi:10.1002/cpa.21747 |
| [28] | Laux, T.; Swartz, D., Convergence of thresholding schemes incorporating bulk effects, Interfaces Free Bound., 19, 273-304 (2017) · Zbl 1376.82004 · doi:10.4171/IFB/383 |
| [29] | Li, H., The volume-preserving mean curvature flow in Euclidean space, Pac. J. Math., 243, 331-355 (2009) · Zbl 1182.53061 · doi:10.2140/pjm.2009.243.331 |
| [30] | Liu, C.; Sato, N.; Tonegawa, Y., On the existence of mean curvature flow with transport term, Interfaces Free Bound, 12, 251-277 (2010) · Zbl 1201.53076 · doi:10.4171/IFB/234 |
| [31] | Luckhaus, S.; Sturzenhecker, T., Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differ. Equ., 3, 253-271 (1995) · Zbl 0821.35003 · doi:10.1007/BF01205007 |
| [32] | Mizuno, M.; Tonegawa, Y., Convergence of the Allen-Cahn equation with Neumann boundary conditions, SIAM J. Math. Anal., 47, 1906-1932 (2015) · Zbl 1330.35196 · doi:10.1137/140987808 |
| [33] | Modica, L.; Mortola, S., Il limite nella \(\Gamma \)-convergenza di una famiglia di funzionali ellittici, Boll. Un. Mat. Ital. A, 14, 526-529 (1977) · Zbl 0364.49006 |
| [34] | Mugnai, L.; Röger, M., The Allen-Cahn action functional in higher dimensions, Interfaces Free Bound, 10, 45-78 (2008) · Zbl 1288.93096 · doi:10.4171/IFB/179 |
| [35] | Mugnai, L.; Seis, C.; Spadaro, E., Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differ. Equ. (2016) · Zbl 1336.53082 · doi:10.1007/s00526-015-0943-x |
| [36] | Pisante, A.; Punzo, F., Allen-Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke’s flows, Commun. Contemp. Math., 17, 1450041 (2015) · Zbl 1336.53083 · doi:10.1142/S0219199714500412 |
| [37] | Pisante, A.; Punzo, F., Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates, Ann. Sc. Norm. Super. Pisa Cl. Sci., 15, 309-341 (2016) · Zbl 1341.53103 |
| [38] | Röger, M.; Schätzle, R., On a modified conjecture of De Giorgi, Math. Z., 254, 675-714 (2006) · Zbl 1126.49010 · doi:10.1007/s00209-006-0002-6 |
| [39] | Rubinstein, J.; Sternberg, P., Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48, 249-264 (1992) · Zbl 0763.35051 · doi:10.1093/imamat/48.3.249 |
| [40] | Simon, L.: Lectures on geometric measure theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, 1983 · Zbl 0546.49019 |
| [41] | Stuvard, S., Tonegawa, Y.: On the existence of canonical multi-phase Brakke flows, arXiv preprint arXiv:2109.14415, 2021 · Zbl 1459.53083 |
| [42] | Takasao, K., Convergence of the Allen-Cahn equation with constraint to Brakke’s mean curvature flow, Adv. Differ. Equ., 22, 765-792 (2017) · Zbl 1479.35531 |
| [43] | Takasao, K., Existence of weak solution for volume preserving mean curvature flow via phase field method, Indiana Univ. Math. J., 66, 2015-2035 (2017) · Zbl 1383.53051 · doi:10.1512/iumj.2017.66.6183 |
| [44] | Takasao, K., On obstacle problem for Brakke’s mean curvature flow, SIAM J. Math. Anal., 53, 6355-6369 (2021) · Zbl 1479.35547 · doi:10.1137/21M1400432 |
| [45] | Takasao, K.; Tonegawa, Y., Existence and regularity of mean curvature flow with transport term in higher dimensions, Math. Ann., 364, 857-935 (2016) · Zbl 1351.53083 · doi:10.1007/s00208-015-1237-5 |
| [46] | Talenti, G., The standard isoperimetric theorem, Handbook of convex geometry, 73-123 (1993), Amsterdam: North-Holland, Amsterdam · Zbl 0799.51015 · doi:10.1016/B978-0-444-89596-7.50008-0 |
| [47] | Tonegawa, Y., Integrality of varifolds in the singular limit of reaction-diffusion equations, Hiroshima Math. J., 33, 323-341 (2003) · Zbl 1059.35061 · doi:10.32917/hmj/1150997978 |
| [48] | Tonegawa, Y., Brakke’s Mean Curvature Flow (2019), Singapore: Springer, Singapore · Zbl 1448.49002 · doi:10.1007/978-981-13-7075-5 |
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