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Remarks on two notions of spectral minimal partitions. (English) Zbl 1222.35067

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)].

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
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