Partitions of Minimal Length on Manifolds

TL;DR

This paper investigates minimal perimeter partitions on 3D manifolds, introducing a relaxed $ ext{Gamma}$-convergence framework, and provides numerical algorithms with results compared to existing literature, especially on spheres.

Contribution

It proposes a novel relaxed framework for minimal perimeter partitions on manifolds and develops an optimization algorithm applicable to general surfaces.

Findings

Numerical results demonstrate the effectiveness of the proposed method.

Comparison with existing literature shows improved or comparable partition quality.

The algorithm provides good approximations starting from relaxed solutions.

Abstract

We study partitions on three dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a $\Gamma$-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.