  The minimal area for surfaces whose border are rectangular and circular loops
are calculated using the Hamilton-Jacobi (HJ) equation. This amounts to solve
the HJ equation for the value of the minimal area, without calculating the
shape of the corresponding surface. This is done for bulk geometries that are
asymptotically AdS. For the rectangular countour, the HJ equation, which is
separable, can be solved exactly. For the circular countour an expansion in
powers of the radius is implemented. The HJ approach naturally leads to a
regularization which consists in locating the countour away from the border.
The results are compared with other regularization which leaves the countour at
the border and calculates the area of the corresponding minimal surface up to a
diameter smaller than the one of the countour at the border. The results do not
coincide, this is traced back to the fact that in the former case the area of a
minimal surface is calculated and in the second the computed area corresponds
to a fraction of a different minimal surface whose countour lies at the
boundary.
