  We show how giant vortices can be stabilized in strong external potential
Bose-Einstein condensates. We illustrate the formation of these vortices thanks
to the relaxation Ginzburg-Landau dynamics for two typical potentials in two
spatial dimensions. The giant vortex stability is studied for the particular
case of the rotating cylindrical hard wall. The minimization of the perturbed
energy is simplified into a one dimensional relaxation dynamics. The giant
vortices can be stabilized only in a finite frequency range. Finally we obtain
a curve for the minimum frequency needed to observe a giant vortex for a given
nonlinearity.
