The Willmore Flow with Small Initial Energy

Kuwert, Ernst; Schätzle, Reiner

doi:10.4310/jdg/1090348128

Cited by 164 publications

(217 citation statements)

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“…Depending on the representation of the evolving surfaces, (1.2) leads to a nonlinear parabolic equation or system which is of fourth order in space. Global existence results for the evolution of closed surfaces can be found in [14], [15]. In addition, two boundary conditions need to be prescribed in case the evolving surfaces have a boundary.…”

Section: Introductionmentioning

confidence: 99%

Error analysis for the approximation of axisymmetric Willmore flow by $C^1$-finite elements

Deckelnick¹,

Schieweck²

2011

Interfaces Free Bound.

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We consider the Willmore flow of axially symmetric surfaces subject to Dirichlet boundary conditions. The corresponding evolution is described by a nonlinear parabolic PDE of fourth order for the radius function. A suitable weak form of the equation, which is based on the first variation of the Willmore energy, leads to a semidiscrete scheme, in which we employ piecewise cubic C 1 -finite elements for the one-dimensional approximation in space. We prove optimal error bounds in Sobolev norms for the solution and its time derivative and present numerical test examples.

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Section: Introductionmentioning

confidence: 99%

Error analysis for the approximation of axisymmetric Willmore flow by $C^1$-finite elements

Deckelnick¹,

Schieweck²

2011

Interfaces Free Bound.

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“…[KuS1,KuS2,KuS3,St]. On the other hand, Mayer and Simonett [MSt] gave a numerical example providing evidence that the Willmore flow may, for particular initial data, develop singularities in finite time.…”

Section: Clamped Plate Modelsmentioning

confidence: 99%

Nonlinear Questions in Clamped Plate Models

Grunau

2009

Milan J. Math.

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Abstract. The linear clamped plate boundary value problem is a classical model in mechanics. The underlying differential equation is elliptic and of fourth order. The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order. Concerning the clamped plate boundary value problem, "linear questions" may be considered as well understood. This changes completely as soon as one poses the simplest "nonlinear question": What can be said about positivity preserving? Does a plate bend upwards when being pushed upwards? It is known that the answer is "no" in general. However, there are many positivity issues as e.g. "almost positivity" to be discussed.Boundary value problems for the "Willmore equation" are nonlinear counterparts for the linear clamped plate equation. The corresponding energy functional involves curvature integrals over the unknown surface. The Willmore equation is of interest in mechanics, membrane physics and, in particular, in differential geometry. Quite far reaching results were achieved concerning closed surfaces. As for boundary value problems, by far less is known. These will be discussed in symmetric situations.

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“…In higher dimension, Kuwert and Schätzle give in [15,16] a long time existence proof of the Willmore flow and the convergence to a round sphere for sufficiently small initial energy. Singularities may appear for larger initial energies, as indicated by numerical simulations [19].…”

Section: On the Willmore Flowmentioning

confidence: 99%