The heat equation shrinking convex plane curves

Gage, Michael E.; Hamilton, Richard S.

doi:10.4310/jdg/1214439902

Cited by 1,000 publications

(796 citation statements)

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“…There is also a volume preserving version of the mean curvature flow, for which one subtracts the average of the mean curvature from the normal velocity. A plethora of analytical results and theories of weak solutions exists, for example, see [20,22,30,35,36,37,43,44,52,53,57,67], and for numerical solutions [1,9,13,14,15,60,69], to name but a few.The algorithm proposed below is a front-tracking boundary-integral method. It has the advantage that one does not have to differentiate across the front, as compared to a level-set approach.…”

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confidence: 99%

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Mayer

2000

Eur. J. Appl. Math

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Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow to treat such free boundary problems in both IR 2 and IR 3 . The advantage of such an approach is the re-usability of much of the setup for all of the different problems.As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity. IntroductionIn this paper we will study geometric evolution problems for surfaces driven by curvature. These moving boundary problems arise from models in physics and the material sciences, and they describe phase changes, or more generally, phenomena in fluid flow.The Hele-Shaw model (named after H.S. ) describes the pressure of two immiscible viscous fluids trapped between two parallel glass plates and has attracted considerable attention in the literature, both on the analytical side [23,27,28,29,33,38,51,54,63] and on the numerical side [2,6,7,8,12,25,26,49,50,65]. This list of references also encompasses work related to the one-sided Hele-Shaw model, which arises as a limit when the viscosity of one of the fluids approaches zero. Furthermore, the Hele-Shaw model has classically been considered on a bounded domain, but many of the references above also address the problem on an unbounded domain. We shall only consider the problem on all of IR n ; for the precise formulation see (4.4).The Mullins-Sekerka model was first proposed by (and much later named after) Mullins and Sekerka to study solidification of materials of negligible specific heat [58]. For a while this model was also called the Hele-Shaw model, leading to a somewhat confused literature. Pego [61], and then Alikakos, Bates, and Chen [3], and also Stoth [68], established this model as a singular limit of the Cahn-Hilliard equation [17], a fourth order partial differential equation modelling nucleation and coarsening phenomena in a melted binary alloy. The Mullins-Sekerka model is considered to be a good model to describe the stage of Ostwald ripening in phase transitions, which is the stage after the initial nucleation has 2 U. F. Mayer essentially been completed and where some particles grow at the cost of others in an effort to decrease interfacial energy (surface area). The literature focuses mainly on the modelling and analytical aspects of the problem [11,16,18,19,21,32,34,46,56,66,71,72] though there are numerical simulations for the two-dimensional case [10,73]. We shall consider the two-phase problem on all of IR n , see (4.3).The mean curvature flow is probably the most studied geometric moving boundary problem, it is in some way also the simplest: the normal velocity is equal to the mean curvature of the interface. This sha...

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confidence: 99%

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Mayer

2000

Eur. J. Appl. Math

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“…The first method to be tried was smoothing by application of the curve shortening flow, [9,11,14], which moves a curve in its normal direction in proportion to its curvature. Curve shortening acts like a nonlinear diffusion equation, smoothing out noise and small-scale features, and eventually contracting a closed curve to a "round point".…”

Section: Smoothingmentioning

confidence: 99%

Automatic Solution of Jigsaw Puzzles

Hoff

Olver

2013

J Math Imaging Vis

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We present a method for automatically solving apictorial jigsaw puzzles that is based on an extension of the method of differential invariant signatures. Our algorithms are designed to solve challenging puzzles, without having to impose any restrictive assumptions on the shape of the puzzle, the shapes of the individual pieces, or their intrinsic arrangement. As a demonstration, the method was successfully used to solve two commercially available puzzles. Finally we perform some preliminary investigations into scalability of the algorithm for even larger puzzles.

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“…Kohn and Serfaty proved that in the case n = 1, involving convex curves in the plane, u is C 3 with D 3 u(x * ) = 0. The analogue of Huisken's work was done for curves in the plane by Gage and Hamilton (see [6]). The regularity of the arrival time was studied in this setting by Kohn and Serfaty in [12].…”

Section: Introductionmentioning

confidence: 99%

“…The regularity of the arrival time was studied in this setting by Kohn and Serfaty in [12]. They needed at least C 3 regularity of u to draw a connection between a minimum exit time of two-person game (see [12] for more details) and the arrival time for a curve shortening flow (see [6]). Their results would completely extend to higher dimensions (drawing a connection between a minimum exit time of the same game as above in higher dimensions and the arrival time of a mean curvature flow) if we knew u were C 3 near x * .…”

Section: Introductionmentioning

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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The heat equation shrinking convex plane curves

Cited by 1,000 publications

References 6 publications

A numerical scheme for moving boundary problems that are gradient flows for the area functional

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Automatic Solution of Jigsaw Puzzles

Rate of convergence of the mean curvature flow

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