Motion of Level Sets by Mean Curvature. I

Evans, Lawrence C.; Spruck, Joel

doi:10.1007/978-3-642-59938-5_13

Cited by 299 publications

(512 citation statements)

References 36 publications

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“…See for instance Bellettini and Novaga [16], [17], Chen, Giga and Goto [20] and Evans and Spruck [26].…”

Section: Theorem 32 (Equivalent Definition For Mean Curvature Type Mmentioning

confidence: 99%

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Lio¹,

Forcadel²,

Monneau³

2008

J. Eur. Math. Soc.

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Abstract. We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.

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“…See for instance Bellettini and Novaga [16], [17], Chen, Giga and Goto [20] and Evans and Spruck [26].…”

Section: Theorem 32 (Equivalent Definition For Mean Curvature Type Mmentioning

confidence: 99%

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Lio¹,

Forcadel²,

Monneau³

2008

J. Eur. Math. Soc.

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“…In order to prove Proposition 6.2, we will use the viscosity solution theory as in [7,10]. Consider the following problem…”

Section: Using the Following Inequality In [15]mentioning

confidence: 99%

Mean curvature flow with a constant forcing

Liu

2012

Nonlinear Differ. Equ. Appl.

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Abstract. In this paper, the motion of surface by its mean curvature and a forcing term θ was studied. We show that to each uniformly convex bounded initial surface M0, there exists a unique θ * such that the surface shrinks or expands depending on whether θ < θ * or θ > θ * . Also, we show that the surface with θ = θ * converges to a limiting surface.Mathematics Subject Classification. 35K57, 58E15.

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“…in Ω and u = 0 at ∂Ω. This equation was first studied by Evans and Spruck in [4]. They showed its solution has the property that each level set u = t is the smooth image of ∂Ω under motion by curvature for time t, for any 0 ≤ t < T .…”

Section: This Contradicts the Maximality Ofmentioning

confidence: 99%

Rate of convergence of the mean curvature flow

Šešum

2007

Comm Pure Appl Math

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We study the flow M t of a smooth, strictly convex hypersurface by its mean curvature in R n+1 . The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x * (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere S n of radius √ n. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us the rate of the exponential decay is at least 2 n . We can define the "arrival time" u of a smooth, strictly convex n-dimensional hypersurface as it moves with normal velocity equal to its mean curvature as u(x) = t, if x ∈ M t for x ∈ Int(M 0 ). Huisken proved that for n ≥ 2 u(x) is C 2 near x * . The case n = 1 has been treated by Kohn and Serfaty, they proved C 3 regularity of u. As a consequence of obtained rate of convergence of the mean curvature flow we prove that u is not C 3 near x * for n ≥ 2. We also show that the obtained rate of convergence 2/n, that comes out from linearizing a mean curvature flow is the optimal one, at least for n ≥ 2.

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Motion of Level Sets by Mean Curvature. I

Cited by 299 publications

References 36 publications

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Mean curvature flow with a constant forcing

Rate of convergence of the mean curvature flow

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