Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

Chen, Yun Gang; Giga, Yoshikazu; Goto, Shun’ichi

doi:10.4310/jdg/1214446564

Cited by 817 publications

(651 citation statements)

References 19 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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“…The collection of sets {rt}t_>0 comprises our generalized evolution by mean curvature. (Chen, Giga, and Goto [4] simultaneously and independently developed a similar approach applied to more general geometric motions. )…”

Section: Introductionmentioning

confidence: 99%

“…Our purpose in this paper is to effect at least a partial reconciliation between the viewpoints of [6,4] and [2] by applying geometric measure theory techniques to analyze the structure of our generalized evolution {Ft}t_>0.…”

Section: Introductionmentioning

confidence: 99%

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Motion of level sets by mean curvature III

1992

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“…This analysis is represented by a viscosity solution [13,16,12] of the following nonlinear degenerate parabolic equation in a two-dimensional rectangular domain…”

Section: Introductionmentioning

confidence: 99%

Solution of nonlinearly curvature driven evolution of plane curves

Mikula

Ševčovič

1999

Applied Numerical Mathematics

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The evolution of plane curves obeying the equation v = β(k), where v is normal velocity and k curvature of the curve is studied. Morphological image and shape multiscale analysis of Alvarez, Guichard, Lions and Morel and affine invariant scale space of curves introduced by Sapiro and Tannenbaum as well as isotropic motions of plane phase interfaces studied by Angenent and Gurtin are included in the model. We introduce and analyze a numerical scheme for solving the governing equation and present numerical experiments.

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Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

Cited by 817 publications

References 19 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

Motion of level sets by mean curvature III

Solution of nonlinearly curvature driven evolution of plane curves

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