3 research outputs found
The ground state of a GrossāPitaevskii energy with general potential in the ThomasāFermi limit
We study the ground state which minimizes a GrossāPitaevskii
energy with general non-radial trapping potential, under the unit mass constraint, in the ThomasāFermi limit where a small parameter tends to 0. This ground state plays an important role in the mathematical treatment of recent
experiments on the phenomenon of BoseāEinstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrodinger equations. Many of these applications require delicate estimates
for the behavior of the ground state near the boundary of the condensate, as the singular parameter tends to zero, 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 superļ¬uid ļ¬ow 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 ThomasāFermi
approximation and accurately approximate the layer by the HastingsāMcLeod solution of the PainleveāII equation. This settles an open problem, answered very recently only for the special case of the model harmonic potential. 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-Holder norm, which is the exact Holder regularity of the singular limit proļ¬le, as the singular parameter tends to zero. Our study is highly motivated by an interesting open problem posed recently by Aftalion, Jerrard, and Royo-Letelier, and an open question of Gallo and Pelinovsky,
concerning the removal of the radial symmetry assumption from the potential trap