An optimal design problem for a charge qubit

In this article, we introduce a simple variational model describing the ground state of a superconducting charge qubit. The model gives rise to a shape optimization problem that aims at maximizing the number of qubit states at a given gating voltage. We show that for small values of the charge, optimal shapes exist and are C^{2, alpha}-nearly spherical sets. In contrast, we prove that balls are not minimizers for large values of the charge and conjecture that optimal shapes do not exist, with the energy favoring disjoint collections of sets.