  We report a series of high-resolution cosmological N-body simulations
designed to explore the formation and properties of dark matter halos with
masses close to the damping scale of the primordial power spectrum of density
fluctuations. We further investigate the effect that the addition of a random
component, v_rms, into the particle velocity field has on the structure of
halos. We adopted as a fiducial model the Lambda Warm Dark Matter cosmology
with a non-thermal sterile neutrino mass of 0.5 keV. The filtering mass
corresponds then to M_f = 2.6x10^12 M_sun/h. Halos of masses close to M_f were
simulated with several million of particles. The results show that, on one
hand, the inner density slope of these halos (at radii <~0.02 the virial radius
Rvir) is systematically steeper than the one corresponding to the NFW fit or to
the CDM counterpart. On the other hand, the overall density profile (radii
larger than 0.02Rvir) is less curved and less concentrated than the NFW fit,
with an outer slope shallower than -3. For simulations with v_rms, the inner
halo density profiles flatten significantly at radii smaller than 2-3 kpc/h
(<~0.010-0.015Rvir). A constant density core is not detected in our
simulations, with the exception of one halo for which the flat core radius is
~1 kpc/h. Nevertheless, if ``cored'' density profiles are used to fit the halo
profiles, the inferred core radii are ~0.1-0.8 kpc/h, in rough agreement with
theoretical predictions based on phase-space constrains, and on dynamical
models of warm gravitational collapse. A reduction of v_rms by a factor of 3
produces a modest decrease in core radii, less than a factor of 1.5. We discuss
the extension of our results into several contexts, for example, to the
structure of the cold DM micro-halos at the damping scale of this model.
