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Communications in Partial Differential Equations Volume 20, 1995 - Issue 11-12
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Original Articles

A Computed example of nonuniqueness of mean curvature flow in R3

S. Angenent Department of Mathematics, University of Wisconsin, Madison, WI, 53706
,
T. Ilmanen Department of Mathematics, University of Wisconsin, Madison, WI, 53706
&
D.L. Chopp Mathematics Department, University of Washington, GN-50, Seattle, WA, 98195
Pages 1937-1958 | Published online: 18 Nov 2011

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David Chopp & JohnA. Veiling. (2003) Foliations of Hyperbolic Space by Constant Mean Curvature Surfaces Sharing Ideal Boundary. Experimental Mathematics 12:3, pages 339-350.
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Yonghoi Koo. (1999) A fattening principle for fronts propagating by mean curvature plus a driving force. Communications in Partial Differential Equations 24:5-6, pages 1035-1053.
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Giovanni Bellettini, Matteo Novaga & Maurizio Paolini. (1998) An example of three dimensional fattening for linked space curves evolving by curvature. Communications in Partial Differential Equations 23:9-10, pages 1475-1492.
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Articles from other publishers (57)

Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis & Felix Schulze. (2024) Mean curvature flow with generic initial data. Inventiones mathematicae.
Crossref
Zhihan Wang. (2024) Mean Convex Smoothing of Mean Convex Cones. Geometric and Functional Analysis 34:1, pages 263-301.
Crossref
Jacob Bernstein & Lu Wang. (2023) An integer degree for asymptotically conical self-expanders. Calculus of Variations and Partial Differential Equations 62:7.
Crossref
John Man Shun Ma, Yuan Shyong Ooi & Juncheol Pyo. (2023) Rigidity results for mean curvature flow graphical translators moving in non-graphical direction. Proceedings of the American Mathematical Society.
Crossref
Junming Xie & Jiangtao Yu. (2023) Convexity of 2-Convex Translating and Expanding Solitons to the Mean Curvature Flow in $${{\mathbb {R}}}^{n+1}$$. The Journal of Geometric Analysis 33:8.
Crossref
Letian Chen. (2023) Rotational symmetry of solutions of mean curvature flow coming out of a double cone II. Calculus of Variations and Partial Differential Equations 62:2.
Crossref
Zhengjiang Lin & Ao Sun. (2021) Bifurcation of perturbations of non-generic closed self-shrinkers. Journal of Topology and Analysis 14:04, pages 979-999.
Crossref
Letian Chen. (2022) Rotational Symmetry of Solutions of Mean Curvature Flow Coming Out of a Double Cone. The Journal of Geometric Analysis 32:10.
Crossref
Jacob Bernstein & Lu Wang. (2022) A Mountain-Pass Theorem for Asymptotically Conical Self-Expanders. Peking Mathematical Journal 5:2, pages 213-278.
Crossref
Sebastian Hensel & Alice Marveggio. (2022) Weak-strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities. Journal of Mathematical Fluid Mechanics 24:3.
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Kyeongsu Choi, Robert Haslhofer, Or Hershkovits & Brian White. (2022) Ancient asymptotically cylindrical flows and applications. Inventiones mathematicae 229:1, pages 139-241.
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Huijuan Wang & Entao Zhao. (2022) A Bernstein type theorem for entire graphic 2-dimensional $\Lambda$-submanifolds in $\mathbf{R}^{4}$. Kodai Mathematical Journal 45:2.
Crossref
Jacob Bernstein & Lu Wang. (2022) Closed hypersurfaces of low entropy in R4 are isotopically trivial. Duke Mathematical Journal 171:7.
Crossref
Jacob Bernstein & Lu Wang. (2022) Relative expander entropy in the presence of a two-sided obstacle and applications. Advances in Mathematics 399, pages 108284.
Crossref
Saul Ancari & Xu Cheng. (2022) Volume properties and rigidity on self-expanders of mean curvature flow. Geometriae Dedicata 216:2.
Crossref
Junfu Yao. (2022) A uniqueness result for self-expanders with small entropy. Proceedings of the American Mathematical Society.
Crossref
Francesco Chini & Niels Martin Møller. (2021) Bi-Halfspace and Convex Hull Theorems for Translating Solitons. International Mathematics Research Notices 2021:17, pages 13011-13045.
Crossref
Jacob Bernstein & Lu Wang. (2021) The space of asymptotically conical self-expanders of mean curvature flow. Mathematische Annalen 380:1-2, pages 175-230.
Crossref
Siao-Hao Guo. (2020) Asymptotic behavior and stability of mean curvature flow with a conical end. Advances in Mathematics 375, pages 107408.
Crossref
Annalisa Cesaroni & Matteo Novaga. (2019) Symmetric Self-Shrinkers for the Fractional Mean Curvature Flow. The Journal of Geometric Analysis 30:4, pages 3698-3715.
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Alix Deruelle & Felix Schulze. (2020) Generic uniqueness of expanders with vanishing relative entropy. Mathematische Annalen 377:3-4, pages 1095-1127.
Crossref
Julian Fischer & Sebastian Hensel. (2020) Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension. Archive for Rational Mechanics and Analysis 236:2, pages 967-1087.
Crossref
Qi Ding. (2019) Minimal cones and self-expanding solutions for mean curvature flows. Mathematische Annalen 376:1-2, pages 359-405.
Crossref
Antonin Chambolle, Massimiliano Morini, Matteo Novaga & Marcello Ponsiglione. (2019) Existence and uniqueness for anisotropic and crystalline mean curvature flows. Journal of the American Mathematical Society 32:3, pages 779-824.
Crossref
Ye Cheng Zhu & Qing Chen. (2019) Gradient Estimates and Harnack Inequalities for Positive Solutions of $$\mathfrak{L}u=\frac{\partial u}{\partial t}$$ on Self-shrinkers. Acta Mathematica Sinica, English Series 35:7, pages 1217-1226.
Crossref
Yecheng Zhu & Qing Chen. (2018) Some integral inequalities for operator and their applications on self-shrinkers . Journal of Mathematical Analysis and Applications 463:2, pages 645-658.
Crossref
Nikolaos Kapouleas, Stephen James Kleene & Niels Martin Møller. (2018) Mean curvature self-shrinkers of high genus: Non-compact examples. Journal für die reine und angewandte Mathematik (Crelles Journal) 2018:739, pages 1-39.
Crossref
Xu Cheng & Detang Zhou. (2018) Spectral properties and rigidity for self-expanding solutions of the mean curvature flows. Mathematische Annalen 371:1-2, pages 371-389.
Crossref
Yecheng Zhu, Yi Fang & Qing Chen. (2017) Complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Science China Mathematics 61:5, pages 929-942.
Crossref
Yecheng Zhu & Qing Chen. (2018) Gradient Estimates for the Positive Solutions of $$\mathfrak {L}u=0$$ L u = 0 on Self-Shrinkers. Mediterranean Journal of Mathematics 15:1.
Crossref
Jiayu Li & Yongbing Zhang. (2017) Lagrangian F-stability of closed Lagrangian self-shrinkers. Journal für die reine und angewandte Mathematik (Crelles Journal) 2017:733, pages 1-23.
Crossref
Antonin Chambolle, Massimiliano Morini & Marcello Ponsiglione. (2016) Existence and Uniqueness for a Crystalline Mean Curvature Flow. Communications on Pure and Applied Mathematics 70:6, pages 1084-1114.
Crossref
Marek Fila, Hiroshi Matano & Eiji Yanagida. 2017. Patterns of Dynamics. Patterns of Dynamics 138 148 .
Yng-Ing Lee & Yang-Kai Lue. (2014) The stability of self-shrinkers of mean curvature flow in higher co-dimension. Transactions of the American Mathematical Society 367:4, pages 2411-2435.
Crossref
Xuan Hien Nguyen. (2014) Construction of complete embedded self-similar surfaces under mean curvature flow, part III. Duke Mathematical Journal 163:11.
Crossref
Sebastian Helmensdorfer & Peter Topping. (2013) Bouncing of charged droplets: An explanation using mean curvature flow. EPL (Europhysics Letters) 104:3, pages 34001.
Crossref
Klaus Ecker. (2012) Partial regularity at the first singular time for hypersurfaces evolving by mean curvature. Mathematische Annalen 356:1, pages 217-240.
Crossref
Hoeskuldur P. Halldorsson. (2012) Helicoidal surfaces rotating/translating under the mean curvature flow. Geometriae Dedicata 162:1, pages 45-65.
Crossref
Tobias Colding & William Minicozzi. (2012) Generic mean curvature flow I; generic singularities. Annals of Mathematics 175:2, pages 755-833.
Crossref
Sebastian Helmensdorfer. (2012) A Model for the Behavior of Fluid Droplets Based on Mean Curvature Flow. SIAM Journal on Mathematical Analysis 44:3, pages 1359-1371.
Crossref
Paweł Biernat & Piotr Bizoń. (2011) Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres. Nonlinearity 24:8, pages 2211-2228.
Crossref
Qing Liu. (2011) Fattening and Comparison Principle for Level-Set Equations of Mean Curvature Type. SIAM Journal on Control and Optimization 49:6, pages 2518-2541.
Crossref
Jasjit S. Suri & Aly A. FaragFuxing Yang, Jasjit S. Suri & Milan Sonka. 2007. Deformable Models. Deformable Models 161 207 .
Henri Anciaux. (2006) Construction of Lagrangian Self-similar Solutions to the Mean Curvature Flow in $$\mathbb{C}^n$$. Geometriae Dedicata 120:1, pages 37-48.
Crossref
Giovanni Bellettini, Vicent Caselles, Antonin Chambolle & Matteo Novaga. (2005) Crystalline Mean Curvature Flow of Convex Sets. Archive for Rational Mechanics and Analysis 179:1, pages 109-152.
Crossref
Michael Feldman, Tom Ilmanen & Lei Ni. (2005) Entropy and reduced distance for Ricci expanders. The Journal of Geometric Analysis 15:1, pages 49-62.
Crossref
Richards Grzhibovskis & Alexei Heintz. (2005) A Convolution-Thresholding Approximation of Generalized Curvature Flows. SIAM Journal on Numerical Analysis 42:6, pages 2652-2670.
Crossref
N.K Yip & P.E Souganidis. (2004) Uniqueness of motion by mean curvature perturbed by stochastic noise. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 21:1, pages 1-23.
Crossref
Andreas Gastel. (2003) Nonuniqueness for the Yang–Mills heat flow. Journal of Differential Equations 187:2, pages 391-411.
Crossref
J.S. Suri, Kecheng Liu, S. Singh, S.N. Laxminarayan, Xiaolan Zeng & L. Reden. (2002) Shape recovery algorithms using level sets in 2-D/3-D medical imagery: a state-of-the-art review. IEEE Transactions on Information Technology in Biomedicine 6:1, pages 8-28.
Crossref
Panagiotis E. Souganidis. 2002. Modern Methods in Scientific Computing and Applications. Modern Methods in Scientific Computing and Applications 397 449 .
Jasjit S. Suri, S. Laxminarayan, Jianbo Gao & Laura Reden. 2002. PDE and Level Sets: Algorithmic Approaches to Static and Motion Imagery. PDE and Level Sets: Algorithmic Approaches to Static and Motion Imagery 153 223 .
Jasjit S. Suri, Sameer Singh & Swamy Laxminarayan. 2002. PDE and Level Sets: Algorithmic Approaches to Static and Motion Imagery. PDE and Level Sets: Algorithmic Approaches to Static and Motion Imagery 97 152 .
J.A. Sethian. (2001) Evolution, Implementation, and Application of Level Set and Fast Marching Methods for Advancing Fronts. Journal of Computational Physics 169:2, pages 503-555.
Crossref
Richard B. Sowers. (1999) Hydrodynamical Limits and Geometric Measure Theory: Mean Curvature Limits from a Threshold Voter Model. Journal of Functional Analysis 169:2, pages 421-455.
Crossref
Martino Bardi, Michael G. Crandall, Lawrence C. Evans, Halil Mete Soner & Panagiotis E. SouganidisPanagiotis E. Souganidis. 1997. Viscosity Solutions and Applications. Viscosity Solutions and Applications 186 242 .
Pedro Martins Girão & Robert V. Kohn. 1995. Variational Methods for Discontinuous Structures. Variational Methods for Discontinuous Structures 7 18 .

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