On spectral minimal partitions II, the case of the rectangle

Abstract

In continuation of \cite{HHOT}, we discuss the question of spectral minimal 3-partitions for the rectangle $]-\frac{a}{2},\frac{a}{2}[\times ]-\frac{b}{2},\frac{b}{2}[$, with $0<a\le b$. It has been observed in \cite{HHOT} that when $0<\frac{a}{b}<\sqrt{\frac{3}{8}}$ the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles $]-\frac{a}{2},\frac{a}{2}[\times ]-\frac{b}{2},-\frac{b}{6}[$, $]-\frac{a}{2},\frac{a}{2}[\times ]-\frac{b}{6},\frac{b}{6}[$ and $]-\frac{a}{2},\frac{a}{2}[\times ]\frac{b}{6},\frac{b}{2}[$. We will describe a possible mechanism of transition for increasing $\frac{a}{b}$ between these nodal minimal 3-partitions and non nodal minimal 3-partitions at the value $\sqrt{\frac{3}{8}}$ and discuss the existence of symmetric candidates for giving minimal 3-partitions when $\sqrt{\frac{3}{8}}<\frac{a}{b}\le 1$. Numerical analysis leads very naturally to nice questions of isospectrality which are solved by introducing Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle.

References in corpus (1)

• Isospectral domains with mixed boundary conditions