Partitions of Minimal Length on Manifolds

ABSTRACT

We study partitions on three-dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a Γ-convergence result and we show some numerical results. We compare our results to those already present in the literature in the case of the sphere. For general surfaces we provide an optimization algorithm on meshes which can give a good approximation of the optimal cost, starting from the results obtained using the relaxed formulation.

KEYWORDS:

• optimal partitions
• gamma convergence
• numerical simulations
• perimeter

2000 AMS SUBJECT CLASSIFICATION:

• 49M20
• 49J45

Acknowledgments

The authors wish to thank Simon Cox and Frank Morgan for useful remarks concerning the manuscript and the software Evolver. They also thank the anonymous reviewer for the useful comments and suggestions concerning this work. The authors gratefully acknowledge the support of the ANR through the projects PGMO and OPTIFORM.

Notes

1 Corresponding to a machine with a 2.2 GHz quad-core processor and 16 GB of RAM.