  In this paper, we consider the initial-boundary value problem to the
nonhomogeneous incompressible Navier-Stokes equations. Local strong solutions
are established, for any initial data $(\rho_0, u_0)\in (W^{1,\gamma} \cap
L^\infty)\times H_{0,\sigma}^1$, with $\gamma>1$, and if $\gamma\geq2$, then
the strong solution is unique. The initial density is allowed to be
nonnegative, and in particular, the initial vacuum is allowed. The assumption
on the initial data is weaker than the previous widely used one that $(\rho_0,
u_0)\in (H^1 \cap L^\infty )\times(H_{0,\sigma}^1 \cap H^2)$, and no
compatibility condition is required.
