Volume 3, Issue 1

We consider closed curves on the sphere moving by the L 2 -gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

For any given C 2 -smooth initial open curves with fixed position and fixed tangent at the boundary points, we obtain the long-time existence of smooth solutions under the second-order evolution of plane curves. Moreover, the asymptotic limit of a convergent subsequence is an inextensible elastica.

We survey some recent results on convex relaxations and a variational approximation for the classical Euclidean Steiner tree problem and we see how these new perspectives can lead to effective numerical schemes for the identification of Steiner minimal trees.

We present some recent results on the possibility of extending the theory of varifolds to the realm of discrete surfaces of any dimension and codimension, for which robust notions of approximate curvatures, also allowing for singularities, can be defined. This framework has applications to discrete and computational geometry, as well as to geometric variational problems in discrete settings. We finally show some numerical tests on point clouds that support and confirm our theoretical findings.

In this paper we review the double covers method with constrained BV functions for solving the classical Plateau’s problem. Next, we carefully analyze some interesting examples of soap films compatible with covers of degree larger than two: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, a soap film that retracts to its boundary, and various soap films spanning an octahedral frame.

Several recent convergence results for numerical schemes for mean-curvature flow in particular in the multi-phase case with arbitrary surface tensions are discussed. The guiding principle of all these works is the gradient-flow structure of multi-phase mean-curvature flow which is explained in the general framework. For simplicity, the convergence results are presented in the simpler two-phase case.

The topology of a minimal cluster of four planar regions with equal areas and smallest possible perimeter was found in [9]. Here we describe the computation used to check that the symmetric cluster with the given topology is indeed the unique minimal cluster.

We summarize in this note a few theoretical and numerical results described with more details in the forthcoming paper [5] devoted to a new phase field model for approximating mean curvature flows. Our work is motivated by the simulation of multiphase mean curvature flows with possibly inhomogeneous surface tensions and mobility coefficients. The key aspect of our model is to incorporate the mobilities not in the phase field energy (which is conventionally the case) but in the metric which determines the gradient flow. We show the consistency of such approach by a formal analysis of the sharp interface limit. We also propose an efficient numerical scheme which allows us to illustrate the advantages of the model on various examples, as the wetting of droplets on solid surfaces or the simulation of nanowires growth generated by the so-called vapor-liquid-solid method.