preprint
Inserted: 21 oct 2026
Last Updated: 22 oct 2026
Year: 2026
AI tools were used in this project. All mathematical arguments and proofs in the final manuscript were checked and written by the author. The author also acknowledges the independent work of Mazzoleni, Moraschi, and Ruffini (arXiv:2610.29958), which proves ball rigidity for the quadratic model by a second shape derivative argument.
Abstract:
We study the shape optimization problem introduced by Mazzoleni, Muratov, and Ruffini as a model for a superconducting charge qubit, in which the first Dirichlet eigenvalue of a set of prescribed volume competes with the Coulomb self-energy of the ground state. We prove that, for sufficiently small charge, the ball is the unique minimizer, and that the energy deficit of a volume-preserving and centered perturbation of the ball is comparable to the squared $H^{1/2}$ norm of the boundary graph, with constants independent of the charge. We also obtain the corresponding results for the quadratic Hartree–Ohta–Kawasaki model.
Download: