  The state of a rotating Bose-Einstein condensate in a harmonic trap is
modeled by a wave function that minimizes the Gross-Pitaevskii functional. The
resulting minimization problem has two new features compared to other similar
functionals arising in condensed matter physics, such as the Ginzburg-Landau
functional. Namely, the wave function is defined in all the plane and is
normalized relative to the $L^2$-norm. This paper deals with the situation when
the coupling constant tends to $0$ (Thomas-Fermi regime) and the rotation speed
is large compared with the first critical speed.
  It is given the leading order estimate of the ground state energy together
with the location of the vortices of the minimizing wave function in the bulk
of the condensate. When the rotation speed is inversely proportional to the
coupling constant, the condensate is confined in an elliptical region whose
conjugate diameter shrinks and whose transverse diameter expands as the
rotation speed increases.
