  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 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 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 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 Painleve-II 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 profile, 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.
