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Stability of spectral partitions with corners

Gregory Berkolaiko, Yaiza Canzani, Graham Cox, Peter Kuchment, Jeremy L. Marzuola

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 $H^{1/2}$.

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 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 .

Paper Structure

This paper contains 30 sections, 31 theorems, 137 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Overview and main results
  3. Definitions and further results
  4. Preliminaries
  5. The two-sided Dirichlet-to-Neumann map
  6. The partition Laplacian
  7. Homology of partitions
  8. Manifolds of diffeomorphisms
  9. Diffeomorphisms of closed manifolds
  10. Boundaries and submanifolds
  11. Variational formulas
  12. Interior variations
  13. Hadamard's formula
  14. Proof of \ref{['interior']}
  15. The second variation of a simple eigenvalue on a $C^3$ domain
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $P$ be a bipartite $k$-partition with corners. The following are equivalent:

Figures (10)

  • Figure 1.1: The so-called "Mercedes star" is conjectured to be the minimal 3-partition of the disk.
  • Figure 1.2: $P$, the partition of the square on the left, locally minimizes $\Lambda$. The partition on the right, $\tilde{P}$, is not included in this local comparison, as it is not of the form $\varphi(P)$ for any diffeomorphism $\varphi$ and hence is not close to $P$. Nonetheless, \ref{['thm:bi']} guarantees that $\Lambda(P) \leq \Lambda(\tilde{P})$.
  • Figure 3.1: Left: a 3-partition of the disk with a ball around each corner point. Right: a magnification showing a covering of part of $\partial M \cup\Sigma$, as in the proof of Lemma \ref{['lem:geo']}. The curve $\gamma$ (arising in step (3)) is shown in red, and the union of the blue and red curves is the set $\Gamma$, the smooth component of $\partial M \cup \Sigma$ that contains $\gamma$.
  • Figure 6.1: The slit disk $D \backslash \Gamma$ (left) and half disk $D_+$ (right) from \ref{['unfold']}.
  • Figure 8.1: The $(2,2)$ partition of the rectangle (left), a qualitative illustration of an energy decreasing deformation (center), and the conjectured topology of a locally minimal 4-partition (right).
  • ...and 5 more figures

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 53 more