  We present a provably stable discontinuous Galerkin spectral element method
for the incompressible Navier-Stokes equations with artificial compressibility
and variable density. Stability proofs, which include boundary conditions, that
follow a continuous entropy analysis are provided. We define a mathematical
entropy function that combines the traditional kinetic energy and an additional
energy term for the artificial compressiblity, and derive its associated
entropy conservation law. The latter allows us to construct a provably stable
split-form nodal Discontinuous Galerkin (DG) approximation that satisfies the
summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The
scheme and the stability proof are presented for general curvilinear
three-dimensional hexahedral meshes. We use the exact Riemann solver and the
Bassi-Rebay 1 (BR1) scheme at the inter-element boundaries for inviscid and
viscous fluxes respectively, and an explicit low storage Runge-Kutta RK3 scheme
to integrate in time. We assess the accuracy and robustness of the method by
solving the Kovasznay flow, the inviscid Taylor-Green vortex, and the
Rayleigh-Taylor instability.
