Stability of spectral partitions with corners

TL;DR

This paper unifies bipartite and non-bipartite spectral partitions on manifolds by introducing a modified Laplacian, linking critical points of an energy functional to eigenfunction nodal partitions, and analyzing their stability and Morse index.

Contribution

It introduces a modified Laplacian operator that characterizes all spectral partitions as eigenfunction nodal sets, extending the understanding of spectral minimal partitions beyond bipartite cases.

Findings

Critical points of the spectral energy functional correspond to nodal partitions of the modified Laplacian.

The Morse index of a critical point equals the nodal deficiency of the eigenfunction.

In bipartite cases, all local minima are global minima; in non-bipartite cases, local minima are topologically constrained.

Abstract

A spectral minimal partition of a manifold is a decomposition into disjoint open sets that minimizes a spectral energy functional. While it is known that bipartite minimal partitions correspond to nodal partitions of Courant-sharp Laplacian eigenfunctions, the non-bipartite case is much more challenging. In this paper, we unify the bipartite and non-bipartite settings by defining a modified Laplacian operator and proving that the nodal partitions of its eigenfunctions are exactly the critical points of the spectral energy functional. Moreover, we prove that the Morse index of a critical point equals the nodal deficiency of the corresponding eigenfunction. Some striking consequences of our main result are: 1) in the bipartite case, every local minimum of the energy functional is in fact a global minimum; 2) in the non-bipartite case, every local minimum of the energy functional minimizes within a certain topological class of partitions. Our results are valid for partitions with non-smooth boundaries; this introduces considerable technical challenges, which are overcome using delicate approximation arguments in the Sobolev space $H^{1/2}$.