Existence and regularity theorems of one-dimensional Brakke flows

Kim, Lami; Tonegawa, Yoshihiro

doi:10.4171/ifb/448

Cited by 10 publications

(12 citation statements)

References 17 publications

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“…). This rescaling was discussed in [19,Section 4], and there exists a subsequence (denoted by the same index) and a limit V = 0 with the properties stated in [19,Theorem 4.1], which shows that V is a unit density varifold and spt V is a real-analytic minimal hypersurface away from a closed lower-dimensional singularity. Moreover, spt V is two-sided in the following sense.…”

Section: Two-sidedness At Unit Density Pointmentioning

confidence: 98%

“…They are improved version of integrality theorem of [18,Theorem 8.6] in that the behaviors of approximating flows and their grains are tracked more in detail. A characterization in [19,Section 4] of limiting behavior within a length scale of o(1/j 2 ), where measure minimizing property dominates, is essentially used. At the end, we describe the proofs of Theorem 2.12 and 2.13.…”

Section: Outline Of the Proofs And Plan Of The Papermentioning

confidence: 99%

“…Appendix A. The existence theorem of [18] revisited Here we point out places which require a change to E vc (E, j) from E(E, j) in [18,19]. It turned out that the proofs require no essential change and the only point to be checked is that the same Lipschitz maps used in the proofs satisfy the condition of Definition 3.1(b).…”

Section: 64)mentioning

confidence: 99%

“…A.4. Changes in [19]. The results from [19] are used in the present paper and the modifications are needed there as well.…”

mentioning

confidence: 99%

“…Changes in [19]. The results from [19] are used in the present paper and the modifications are needed there as well. On the other hand, similarly, one can check that the change to the volume-controlled counterpart does not cause any difficulties.…”

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confidence: 99%

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On the existence of canonical multi-phase Brakke flows

Stuvard,

Tonegawa

2021

Preprint

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This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. Under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows. Contents 1. Introduction 2. Notation and main results 2.1. Basic notation 2.2. Three weak notions of MCF 2.3. Main results 2.4. Outline of the proofs and plan of the paper 2.5. Further notation 3. Existence of multi-phase Brakke flow 3.1. Volume-controlled Lipschitz deformations 3.2. The constructive scheme 4. BV flow: proof of Theorem 2.11(6) 4.1. Characterization as L 2 flow 4.2. Existence of measure-theoretic velocities 4.3. Boundaries move by their mean curvature 5. Ilmanen's density lower bound 6. Two-sidedness at unit density point 7. Final remarks 7.1. On generalized BV solutions 7.2. MCF with fixed boundary conditions 7.3. A lower bound estimate for extinction time Appendix A. The existence theorem of [18] revisited A.1. Construction of approximate flows A.2. Proofs of rectifiability and integrality A.3. Volume change of grains A.4. Changes in [19] References

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Section: Two-sidedness At Unit Density Pointmentioning

confidence: 98%

Section: Outline Of the Proofs And Plan Of The Papermentioning

confidence: 99%

Section: 64)mentioning

confidence: 99%

“…A.4. Changes in [19]. The results from [19] are used in the present paper and the modifications are needed there as well.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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On the existence of canonical multi-phase Brakke flows

Stuvard,

Tonegawa

2021

Preprint

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Short‐time existence for the network flow

Lira

Mazzeo

Pluda

et al. 2023

Comm Pure Appl Math

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This paper contains a new proof of the short‐time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard‐Reitich, Mantegazza‐Novaga‐Tortorelli and eventually, in the most general case of irregular networks by Ilmanen‐Neves‐Schulze. Although the present paper proves a result similar to the one in Ilmanen et al., the method here provides substantially more detailed information about how an irregular network “resolves” into a regular one. Either approach relies on the existence of self‐similar expanding solutions found in Mazzeo and Saez. As a precursor to the main theorem, we also prove an unexpected regularity result for the mixed Cauchy‐Dirichlet boundary problem for the linear heat equation on a manifold with boundary.

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Łojasiewicz–Simon inequalities for minimal networks: stability and convergence

Pluda,

Pozzetta

2023

Math. Ann.

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We investigate stability properties of the motion by curvature of planar networks. We prove Łojasiewicz–Simon gradient inequalities for the length functional of planar networks with triple junctions. In particular, such an inequality holds for networks with junctions forming angles equal to $$\tfrac{2}{3}\pi $$ 2 3 π that are close in $$H^2$$ H 2 -norm to minimal networks, i.e., networks whose edges also have vanishing curvature. The latter inequality bounds a concave power of the difference between length of a minimal network $$\Gamma _*$$ Γ ∗ and length of a triple junctions network $$\Gamma $$ Γ from above by the $$L^2$$ L 2 -norm of the curvature of the edges of $$\Gamma $$ Γ . We apply this result to prove the stability of minimal networks in the sense that a motion by curvature starting from a network sufficiently close in $$H^2$$ H 2 -norm to a minimal one exists for all times and smoothly converges. We further rigorously construct an example of a motion by curvature having uniformly bounded curvature that smoothly converges to a degenerate network in infinite time.

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Existence and regularity theorems of one-dimensional Brakke flows

Cited by 10 publications

References 17 publications

On the existence of canonical multi-phase Brakke flows

On the existence of canonical multi-phase Brakke flows

Short‐time existence for the network flow

Łojasiewicz–Simon inequalities for minimal networks: stability and convergence

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