Dynamics of dark energy in the gravitational fields of matter
  inhomogeneities
  We study the dynamical properties and space distribution of dark energy in
the weak and strong gravitational fields caused by inhomogeneities of matter in
the static world of galaxies and clusters. We show that the dark energy in the
weak gravitational fields of matter density perturbations can condense or
dilute, but amplitudes of its perturbations remain very small on all scales. We
illustrate also how the "accretion" of the phantom dark energy onto the matter
overdensity forms the dark energy underdensity. We analyze the behavior of dark
energy in the gravitational fields of stars and black holes with the
Schwarzschild metric. It is shown that, in the case of stars, the static
solution of the differential equations for energy-momentum conservation exists
and describes the distribution of density of dark energy inside and outside of
a star. We have found that for stars and galaxies its value differs slightly
from the average and is a bit higher for the quintessential scalar field as
dark energy and a bit lower for the phantom one. The difference grows with the
decrease of the effective sound speed of dark energy and is large in the
neighborhood of neutron stars. We obtain and analyze also the solutions of
equations that describe the stationary accretion of the dark energy as a test
component onto the Schwarzschild black hole. It is shown that the rate of
change of mass of the dark energy is positive in the case of quintessential
dark energy and is negative in the case of the phantom one.
