Global solutions to the volume-preserving mean-curvature flow

Mugnai, Luca; Seis, Christian; Spadaro, Emanuele

doi:10.1007/s00526-015-0943-x

Cited by 36 publications

(74 citation statements)

References 35 publications

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“…They prove (conditioned) convergence of a scheme introduced by Ruuth and Wetton in [35]. We also draw the reader's attention to the recent work of Mugnai et al [32], where they prove a (conditional) convergence result as in [24] of a modification of the scheme in [2,24] to volume-preserving mean curvature flow. Note that due to the only conditional convergence, our result does not provide a long-time existence result for (weak solutions of) multi-phase mean curvature flow.…”

Section: Contextmentioning

confidence: 86%

Convergence of the thresholding scheme for multi-phase mean-curvature flow

2016

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We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman et al. (Diffusion generated motion by mean curvature. Department of Mathematics, University of California, Los Angeles 1992). We prove a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation of mean curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movements scheme by Esedoglu et al. (Commun Pure Appl Math 68(5):808-864, 2015). This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean curvature flow as a gradient flow w. r. t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movements scheme for an energy functional that -converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren et al. (SIAM J Control Optim 31(2):387-438, 1993) and Luckhaus and Sturzenhecker (Calculus Var Partial Differ Equ 3(2):253-271, 1995), which establish convergence of a more academic minimizing movements scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which however is not ensured by the compactness coming from the basic estimates.

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Section: Contextmentioning

confidence: 86%

Convergence of the thresholding scheme for multi-phase mean-curvature flow

2016

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“…Before passing to the next step let us define Step 4: Uniform density estimates for minimizers. Given λ > C 5 and a minimizer E λ of F 2 (·, E 0 , λ), following arguments of [42,48] let us show that…”

Section: Existence Of Gmm For Bounded Partitionsmentioning

confidence: 99%

“…Step 5: Existence of GMM starting from G . We follow the arguments of [42,48]. Let {G(λ, k)} λ>C 5 ,k∈N 0 be defined as follows: G(λ, 0) = G and G(λ, k) ∈ argmin F 2 (·, G(λ, k − 1), λ), k ≥ 1.…”

Section: Existence Of Gmm For Bounded Partitionsmentioning

confidence: 99%

Minimizing Movements for Mean Curvature Flow of Partitions

Bellettini¹,

Kholmatov²

2018

SIAM J. Math. Anal.

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Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1 n+1 -Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1 2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

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“…In section 3 we prove Theorem 1 and Corollary 2. The generalized Heintze-Karcher inequality for sets of finite perimeter is stated and proved in section 4. ing out to us, respectively, the references [Wen86] and [MSS16]. Part of this work was completed while both authors were affiliated to the Abdus Salam International Centre for Theoretical Physics in Trieste, Italy.…”

mentioning

confidence: 99%