The Ground State of a Gross–pitaevskii Energy with General Potential in the Thomas–fermi Limit

We study the ground state which minimizes a Gross–Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the Thomas–Fermi limit where a small parameter ε tends to 0. This ground state plays an important role in the mathematical treatment of recent experiments on the phenomenon of Bose–Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrödinger equations. Many of these applications require delicate estimates for the behavior of the ground state near the boundary of the condensate, as ε → 0, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the condensate, understand the superfluid flow around an obstacle, and also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the classical Thomas–Fermi approximation and accurately approximate the layer by the Hastings–McLeod solution of the Painlevé–II equation. This settles an open problem (cf. [9, pg. 13 or Open Problem 8.1]), answered very recently only for the special case of the model harmonic potential [104]. In fact, we even improve upon previous results that relied heavily on the radial symmetry of the potential trap. Moreover, we show that the ground state has the maximal regularity available, namely it remains uniformly bounded in the 1 2 -Hölder norm, which is the exact Hölder regularity of the singular limit profile, as ε → 0. Our study is highly motivated by an interesting open problem posed recently by Aftalion, Jerrard, and Royo-Letelier [10], and an open question of Gallo and Pelinovsky [103], concerning the removal of the radial symmetry assumption from the potential trap.