Geometric Evolution of Phase-Boundaries

Giga, Yoshikazu; Goto, Shun’ichi

doi:10.1007/978-1-4613-9211-8_3

Cited by 23 publications

(27 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Evans and Spruck [15], [16], Chen, Giga and Goto [6], [7] work with hypersurfaces which are defined as level sets of viscosity solutions of a nonlinear partial differential equation on some domain in R n+1 . Regularity results are given in [15], [16], [18], [25], [26].…”

Section: (P T) = −(H(p T) − H(t))ν(p T) P ∈mentioning

confidence: 99%

Volume-preserving mean curvature flow of rotationally symmetric surfaces

Athanassenas¹

1997

Comment. Math. Helv.

View full text Add to dashboard Cite

Abstract.A rotationally symmetric n-dimensional surface in R n+1 , of enclosed volume V and with boundary in two parallel planes, is evolving under volume-preserving mean curvature flow. For large volume V , we obtain gradient and curvature estimates, leading to long-time existence of the flow, and convergence to a constant mean curvature surface.Mathematics Subject Classification (1991). 35K22, 58G11, 53C21.

show abstract

Section: (P T) = −(H(p T) − H(t))ν(p T) P ∈mentioning

confidence: 99%

Volume-preserving mean curvature flow of rotationally symmetric surfaces

Athanassenas¹

1997

Comment. Math. Helv.

View full text Add to dashboard Cite

show abstract

“…The methods of [Gerhardt 2006, Section 2.5] (see also [Giga and Goto 1992] and [Baker 2010]) then imply short time existence of solutions, so long as the principal curvatures of the initial immersion lie in .…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

Convexity estimates for hypersurfaces moving by convex curvature functions

2014

View full text Add to dashboard Cite

We consider the evolution of compact hypersurfaces by fully non-linear, parabolic curvature ows for which the normal speed is given by a smooth, convex, degree one homoge-neous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the ow. The result extends the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature ow to a large class of speeds, and leads to an analogous description of `type-II' singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data. We consider the evolution of compact hypersurfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, convex, degree-one homogeneous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the flow. The result extends the convexity estimate of Huisken and Sinestrari [Acta Math. 183:1 (1999), 45-70] for the mean curvature flow to a large class of speeds, and leads to an analogous description of "type-II" singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data.

show abstract

“…The proof is an adaptation of the one proposed by Evans and Spruck [16] for the classical mean curvature motion (see also Giga and Goto [20] and Maekawa [24] for more general equations). For the reader's convenience, we give the steps of the proof to explain how to treat the dependence in the space variable of the velocity c.…”

Section: Existence and Uniqueness Of A Smooth Solutionmentioning

confidence: 90%

Minimizing movements for dislocation dynamics with a mean curvature term

Forcadel¹,

Monteillet²

2009

ESAIM: COCV

View full text Add to dashboard Cite

Abstract. We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.Mathematics Subject Classification. 53C44, 49Q15, 49L25, 28A75, 58A25.

show abstract

Geometric Evolution of Phase-Boundaries

Cited by 23 publications

References 13 publications

Volume-preserving mean curvature flow of rotationally symmetric surfaces

Volume-preserving mean curvature flow of rotationally symmetric surfaces

Convexity estimates for hypersurfaces moving by convex curvature functions

Minimizing movements for dislocation dynamics with a mean curvature term

Contact Info

Product

Resources

About