Communications in Mathematical Sciences

Volume 12 (2014)

Number 1

Colliding interfaces in old and new diffuse-interface approximations of Willmore-flow

Pages: 125 – 147

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n1.a6

Authors

Selim Esedoglu (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Andreas Rätz (Department of Mathematics, TU Dortmund, Germany)

Matthias Röger (Department of Mathematics, TU Dortmund, Germany)

Abstract

This paper is concerned with difuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the $L_1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation.

Keywords

diffuse-interface, Willmore flow, elastica energy, topological change

2010 Mathematics Subject Classification

49J45, 53C44, 74S05

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