Volume preserving mean curvature flow for star-shaped sets

Kim, In-Won; Kwon, Dohyun

doi:10.1007/s00526-020-01738-0

Cited by 20 publications

(17 citation statements)

References 20 publications

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“…One can also consider initial data for which topological changes do not occur like star-shaped sets in the isotropic case [KK20] or sets that satisfy a certain reflection symmetry property in the anisotropic case including some crystalline flow [KKP]. 9.2.…”

Section: Some Numericsmentioning

confidence: 99%

Motion by crystalline-like mean curvature: a survey

Giga¹,

Požár²

2021

Preprint

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We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem. Contents 1. Introduction 2. Some models 3. Polygonal flow 4. Explicit solutions 5. Approach by the theory of maximal monotone operators 5.1. Abstract theory 5.2. Calibrability and Cheeger sets 5.3. Curvature-like quantity 5.4. Comparison and approximation 6. Approach by the theory of viscosity solutions 6.1. Definition of viscosity solutions 6.2. Comparison principle 6.3. Existence of solutions 6.4. Convergence of various approximations 7. Approach by distance functions 8. Some numerics 9. Volume-preserving and fourth-order problems 9.1. Volume preserving flow 9.2. Fourth-order problem References

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Section: Some Numericsmentioning

confidence: 99%

Motion by crystalline-like mean curvature: a survey

Giga¹,

Požár²

2021

Preprint

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“…For example one can consider uniformly convex and nearly spherical initial sets (see [9,10]), or C ∞ −regular initial sets that are H 3 −close to strictly stable critical sets in the three and four dimensional flat torus (see [22]). For more general initial data, the long time behaviour in the context of flat flows of convex and star-shaped sets (see [5,12]) has been characterized only up to (possibly diverging in the case of [5]) translations. In [21] the authors characterized the long-time limits of the discrete-in-time approximate flows constructed by the Euler implicit scheme introduced in [2,14] under the volume constraint in arbitrary space dimension.…”

Section: Introductionmentioning

confidence: 99%

Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

De Gennaro,

Kubin

2022

Preprint

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We show that the discrete approximate volume preserving mean curvature flow in the flat torus T N starting near a strictly stable critical set E of the perimeter converges in the long time to a translate of E exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.

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“…There are several asymptotic analysis results on the forced mean curvature flows with Neumann boundary conditions [13,21,22,24] or with periodic boundary conditions [3], but they are all for graph-like surfaces. The volume preserving mean curvature flow, which is a different type of forced mean curvature flows, was studied in [17,18]. Recently, the relation between the level set approach and the varifold approach for (1.1) with c ≡ 0 was investigated in [1].…”

Section: Introductionmentioning

confidence: 99%

Level-set forced mean curvature flow with the Neumann boundary condition

Jang¹,

Kwon²,

Mitake³

et al. 2021

Preprint

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Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

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Volume preserving mean curvature flow for star-shaped sets

Cited by 20 publications

References 20 publications

Motion by crystalline-like mean curvature: a survey

Motion by crystalline-like mean curvature: a survey

Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

Level-set forced mean curvature flow with the Neumann boundary condition

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