Stationary sets of the mean curvature flow with a forcing term
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Vesa Julin
und
Joonas Niinikoski
Abstract
We consider the flat flow solution to the mean curvature equation with forcing in
Funding source: Academy of Finland
Award Identifier / Grant number: 314227
Funding statement: The research was supported by the Academy of Finland grant 314227.
References
[1] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: A variational approach, SIAM J. Control Optim. 31 (1993), no. 2, 387–438. 10.1137/0331020Suche in Google Scholar
[2] M. Barchiesi, F. Cagnetti and N. Fusco, Stability of the Steiner symmetrization of convex sets, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1245–1278. 10.4171/jems/391Suche in Google Scholar
[3] G. Bellettini, Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations, Appunti. Sc. Norm. Super. Pisa (N. S.) 12, Edizioni della Normale, Pisa, 2013. 10.1007/978-88-7642-429-8Suche in Google Scholar
[4] G. Bellettini and M. Paolini, Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 43–67; Erratum, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 26 (2002), 161–165. Suche in Google Scholar
[5] K. A. Brakke, The Motion of a Surface By its Mean Curvature, Math. Notes 20, Princeton University, Princeton, 1978. Suche in Google Scholar
[6] 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-zSuche in Google Scholar
[7] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 207–210. 10.3792/pjaa.65.207Suche in Google Scholar
[8] M. G. Delgadino and F. Maggi, Alexandrov’s theorem revisited, Anal. PDE 12 (2019), no. 6, 1613–1642. 10.2140/apde.2019.12.1613Suche in Google Scholar
[9] N. Dirr, S. Luckhaus and M. Novaga, A stochastic selection principle in case of fattening for curvature flow, Calc. Var. Partial Differential Equations 13 (2001), no. 4, 405–425. 10.1007/s005260100080Suche in Google Scholar
[10] 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/1214446559Suche in Google Scholar
[11] N. Fusco, V. Julin and M. Morini, Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, preprint (2020), https://arxiv.org/abs/2004.07734. Suche in Google Scholar
[12] M. A. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J. 58 (1989), no. 3, 555–558. 10.1215/S0012-7094-89-05825-0Suche in Google Scholar
[13] R. Gulliver and Y. Koo, Sharp growth rate for generalized solutions evolving by mean curvature plus a forcing term, J. Reine Angew. Math. 538 (2001), 1–24. 10.1515/crll.2001.065Suche in Google Scholar
[14] 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/BF01205007Suche in Google Scholar
[15] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Stud. Adv. Math. 135, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139108133Suche in Google Scholar
[16] 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-xSuche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Intrinsic scaling method for doubly nonlinear parabolic equations and its application
- Causal variational principles in the infinite-dimensional setting: Existence of minimizers
- A Li–Yau inequality for the 1-dimensional Willmore energy
- BV and Sobolev homeomorphisms between metric measure spaces and the plane
- HW2,2 loc-regularity for p-harmonic functions in Heisenberg groups
- Stationary sets of the mean curvature flow with a forcing term
- Approximation of the Willmore energy by a discrete geometry model
- Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian
- Lipschitz regularity for degenerate elliptic integrals with 𝑝, 𝑞-growth
- Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems
- Minimality of balls in the small volume regime for a general Gamow-type functional
- Dimension estimates for the boundary of planar Sobolev extension domains
Artikel in diesem Heft
- Frontmatter
- Intrinsic scaling method for doubly nonlinear parabolic equations and its application
- Causal variational principles in the infinite-dimensional setting: Existence of minimizers
- A Li–Yau inequality for the 1-dimensional Willmore energy
- BV and Sobolev homeomorphisms between metric measure spaces and the plane
- HW2,2 loc-regularity for p-harmonic functions in Heisenberg groups
- Stationary sets of the mean curvature flow with a forcing term
- Approximation of the Willmore energy by a discrete geometry model
- Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian
- Lipschitz regularity for degenerate elliptic integrals with 𝑝, 𝑞-growth
- Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems
- Minimality of balls in the small volume regime for a general Gamow-type functional
- Dimension estimates for the boundary of planar Sobolev extension domains
