In this talk I shall recall a quite famous variational energy from quantum mechanics: the Gamow [liquid drop] model, and some questions about it (some solved, some open). Then I will consider two more energies: the first is the counterpart of the Gamow model with a spectral energy term instead of a surface energy one. The second is a reduced Hartree type energy. These energies have very similar behavior and quite similar mathematical formulations. For the latter two I will present some optimal design problems. Precisely, in some regimes one can show that the ball is a rigid minimizer. I will outline the strategy to do that for one of the two cases. The talk is based on the paper [MR] in collaboration with D. Mazzoleni (Pavia) and on a work in progress in collaboration with D. Mazzoleni and C. B. Muratov (Pisa).

[MR] Mazzoleni, Dario (I-PAVI); Ruffini, Berardo (I-BOLO): A spectral shape optimization problem with a nonlocal competing term. (English summary) Calc. Var. Partial Differential Equations 60 (2021), no. 3, Paper No. 114, 46 pp.