  We study a singular limit of the classical parabolic-elliptic
Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The
main result is the $\Gamma$ convergence of the corresponding energy functional
toward the perimeter functional. Following recent work on this topic, we then
prove that under an energy convergence assumption, the solution of the PKS
model converges to a solution of the Hele-Shaw free boundary problem with
surface tension, which describes the evolution of the interface separating
regions with high density from those with low density. This result complements
a recent work by the author with I. Kim and Y. Wu, in which the same free
boundary problem is derived from the incompressible PKS model (which includes a
density constraint $\rho\leq 1$ and a pressure term): It shows that the
incompressibility constraint is not necessary to observe phase separation and
surface tension phenomena.
