Helffer, Bernard; Hoffmann-Ostenhof, Thomas Remarks on two notions of spectral minimal partitions. (English) Zbl 1222.35067 Adv. Math. Sci. Appl. 20, No. 1, 249-263 (2010). Summary: In continuation of previous work [the authors and S. Terracini, Ann. Inst. H. Poincaré, Anal. Non Linéaire 26, 101–138 (2009; Zbl 1171.35083); “Nodal domains and spectral minimal partitions. The case of the sphere” (to appear)], we analyze the properties of spectral minimal partitions. We focus on the comparison between two definitions of minimal partitions and give some simple rather generic criterion implying that they cannot coincide. We illustrate this criterion in the case of simple convex examples like the rectangle and the equilateral triangle. This partially answers a question posed by D. Bucur, G. Buttazzo and A. Henrot [Adv. Math. Sci. Appl. 8, No. 2, 571–579 (1998; Zbl 0915.49006)]. Cited in 6 Documents MSC: 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35P15 Estimates of eigenvalues in context of PDEs Keywords:spectral minimal partitions Citations:Zbl 1171.35083; Zbl 0915.49006 PDFBibTeX XMLCite \textit{B. Helffer} and \textit{T. Hoffmann-Ostenhof}, Adv. Math. Sci. Appl. 20, No. 1, 249--263 (2010; Zbl 1222.35067)