Remarks on the boundary set of spectral equipartitions

Abstract : Given a bounded open set $\Omega$ in $\mathbb{R}^n$ (or a compact Riemannian manifold with boundary), and a partition of $\Omega$ by $k$ open sets $\omega_j$, we consider the quantity $\max_j \lambda(\omega_j)$, where $\lambda(\omega_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $\omega_j$. We denote by $\mathfrak{L}_k(\Omega)$ the infimum of $\max_j \lambda(\omega_j)$ over all $k$-partitions. A minimal $k$-partition is a partition which realizes the infimum. The purpose of this paper is to revisit properties of nodal sets and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We focus on the length of the boundary set of the partition in the $2$-dimensional situation.
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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 372, pp.20120492. <http://rsta.royalsocietypublishing.org/>. <10.1098/rsta.2012.0492>
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Pierre Bérard, Bernard Helffer. Remarks on the boundary set of spectral equipartitions. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 372, pp.20120492. <http://rsta.royalsocietypublishing.org/>. <10.1098/rsta.2012.0492>. <hal-00678905v2>

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