Homology of spectral minimal partitions

TL;DR

This paper explores the topology of spectral minimal partitions on manifolds, introducing a modified Laplacian to analyze non-bipartite cases and applying algebraic topology tools to classify partition types.

Contribution

It introduces a modified Laplacian operator and uses algebraic topology to characterize and analyze non-bipartite spectral minimal partitions.

Findings

Nodal partitions of Courant-sharp eigenfunctions are minimal within certain topological classes.

The approach recovers known bipartite results and extends understanding to non-bipartite partitions.

Topological types of partitions are characterized by relative homology.

Abstract

A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant-sharp Laplacian eigenfunctions. However, almost all minimal partitions are non-bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant-sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non-bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology.