Stability of spectral partitions with corners

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 nonbipartite case is much more challenging. In this paper, we unify the bipartite and nonbipartite 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 nonbipartite case, every local minimum of the energy functional minimizes within a certain topological class of partitions. Our results are valid for partitions with nonsmooth boundaries; this introduces considerable technical challenges, which are overcome using delicate approximation arguments in the Sobolev space $H^{1/2}$.