On spectral minimal partitions II, the case of the rectangle

Virginie Bonnaillie-Noël (1) , Bernard Helffer (2) , Thomas Hoffmann-Ostenhof (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 Erwin Schrödinger Institute for Mathematical Physics

In continuation of \cite{HHOT}, we discuss the question of spectral minimal 3-partitions for the rectangle $]-a/2,a/2[\times ]-b/2,b/2[\,$, with $0< a\leq b$. It has been observed in \cite{HHOT} that when $0< a/b < \sqrt{3/8}$ the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles $]-a/2,a/2[\times ]-b/2,-b/6[$, $]-a/2,a/2[\times ]-b/6,b/6[$ and $]-a/2,a/2[\times ]b/6,b/2[$. We will describe a possible mechanism of transition for increasing $a/b$ between these nodal minimal 3-partitions and non nodal minimal 3-partitions at the value $\sqrt{3/8}$ and discuss the existence of symmetric candidates for giving minimal 3-partitions when $\sqrt{3/8}

Mots clés: Spectral theory; minimal partition; nodal set; Aharonov-Bohm Hamiltonian; isospectrality; finite method for problems with mixed boundary conditions

Domaines: Analyse numérique [math.NA]

Virginie Bonnaillie-Noël, Bernard Helffer, Thomas Hoffmann-Ostenhof. On spectral minimal partitions II, the case of the rectangle. 2008. ⟨hal-00323849⟩