A Convex Approach to Minimal Partitions

We describe a convex relaxation for a family of problems of minimal perimeter partitions. The minimization of the relaxed problem can be tackled numerically, we describe an algorithm and show some results. In most cases, our relaxed problem finds a correct numerical approximation of the optimal solution: we give some arguments to explain why it should be so, and also discuss some situation where it fails. This preprint is a revised version of an technical paper of 2008 (see for instance the CMAP preprint #649, november 2008), which was rewritten in order to clarify, in particular, the relationship with the classical "paired calibration" approach for minimal surfaces