Generalized Motion¶by Nonlocal Curvature in the Plane

Giga, Mi-Ho; Giga, Yoshikazu

doi:10.1007/s002050100154

Cited by 60 publications

(95 citation statements)

References 32 publications

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“…This is a very special version of results in [GG4] or [GG5], where level set equations to more general equation of the form V = g(n, −div Γt (n)) is studied. The main idea of the proof is to reduce the problem to graph-like solutions of (2.1) which is studied in [GG2] and [GG3].…”

Section: Applicationmentioning

confidence: 95%

“…[CGG1], [Ca], and [GG5]). However, if do not assume the convergence of derivatives of γ τ , it is quite recent that the convergence ansatz has been proved for n = 2 in [GG4] and [GG5]. Note that meaning of a solution to (2.4) is not clear at all for all nondifferentiable γ since the term divξ(∇u) is not well-defined even for smooth u.…”

Section: Applicationmentioning

confidence: 99%

“…Note that meaning of a solution to (2.4) is not clear at all for all nondifferentiable γ since the term divξ(∇u) is not well-defined even for smooth u. The papers [GG4] and [GG5] provide a proper notion of the solution.…”

Section: Applicationmentioning

confidence: 99%

“…No control of derivatives of γ is necessary. This gives a way to approximate crystalline motion [T], [AG] in the plane by an anisotropic Allen-Cahn type equation in conjunction with a general level set method for nondifferentiable γ in [GG4], [GG5]. This will be explained in §2.5 as an application of our main result.…”

Section: Introductionmentioning

confidence: 98%

See 3 more Smart Citations

On a uniform approximation of motion by anisotropic curvature by the Allen–Cahn equations

Giga¹,

Ohtsuka²,

Schätzle³

2006

Interfaces Free Bound.

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The convergence of solutions of anisotropic Allen-Cahn equations is studied when the interface thickness parameter (denoted by ε) tends to zero. It is shown that the convergence to a level set solution of the corresponding anisotropic interface equations is uniform with respect to the derivatives of a surface energy density function. As an application a crystalline motion of interfaces is shown to be approximated by anisotropic Allen-Cahn equations.2000 Mathematics Subject Classification: 35B25, 35K57, 53C44.

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Section: Applicationmentioning

confidence: 95%

Section: Applicationmentioning

confidence: 99%

Section: Applicationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

On a uniform approximation of motion by anisotropic curvature by the Allen–Cahn equations

Giga¹,

Ohtsuka²,

Schätzle³

2006

Interfaces Free Bound.

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“…In particular, we are interested in examples of facets F ⊂ ∂E which admit a subunitary vector field allowing to define an anisotropic mean curvature not easily expressible in terms of a scalar function. The study of anisotropic mean curvature of facets is related to crystalline mean curvature flow [68], [70], [71], [2], [48], [49]: for instance, the constancy of the crystalline mean curvature makes a facet translate parallely to itself in normal direction, at least for a short time, thus preventing the facet-breaking and bending phenomena [21].…”

Section: Introductionmentioning

confidence: 99%

Anisotropic mean curvature on facets and relations with capillarity

Amato¹,

Bellettini²,

Tealdi³

2015

Geometric Flows

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Abstract:Given an anisotropy ϕ on R 3 , we discuss the relations between the ϕ-calibrability of a facet F ⊂ ∂E of a solid crystal E, and the capillary problem on a capillary tube with base F. When F is parallel to a facet︀ B F ϕ of the unit ball of ϕ, ϕ-calibrability is equivalent to show the existence of a ϕ-subunitary vector field in F, with suitable normal trace on ∂F, and with constant divergence equal to the ϕ-mean curvature of F. Assuming E convex at F,̃︀ B F ϕ a disk, and F (strictly) ϕ-calibrable, such a vector field is obtained by solving the capillary problem on F in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict ϕ-calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C 1,1 , the solution of the total variation flow starting at 1 F .

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Existence and Uniqueness for a Crystalline Mean Curvature Flow

Chambolle

Morini

Ponsiglione

2016

Comm Pure Appl Math

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Abstract. An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The comparison principle is obtained by means of a suitable weak formulation of the flow, while the existence of a global-in-time solution follows via a minimizing movements approach.

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Generalized Motion¶by Nonlocal Curvature in the Plane

Cited by 60 publications

References 32 publications

On a uniform approximation of motion by anisotropic curvature by the Allen–Cahn equations

On a uniform approximation of motion by anisotropic curvature by the Allen–Cahn equations

Anisotropic mean curvature on facets and relations with capillarity

Existence and Uniqueness for a Crystalline Mean Curvature Flow

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