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.
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