On the faint-end of the galaxy luminosity function in the Epoch of
  Reionization: updated constraints from the HST Frontier Fields
  Ultra-faint galaxies are hosted by small dark matter halos with shallow
gravitational potential wells, hence their star formation activity is more
sensitive to feedback effects. The shape of the faint-end of the high-$z$
galaxy luminosity function (LF) contains important information on star
formation and its interaction with the reionization process during the Epoch of
Reionization (EoR). High-$z$ galaxies with $M_{\rm UV}\gtrsim-17$ have only
recently become accessible thanks to the Frontier Fields (FFs) survey combining
deep {\it HST} imaging and the gravitational lensing effect. In this paper we
investigate the faint-end of the LF at redshift $>$5 using the data of FFs
clusters Abell 2744 (A2744), MACSJ0416.1-2403 (M0416), MACSJ0717.5+3745 (M0717)
and MACSJ1149.5+2223 (M1149). We analyze both an empirical and a
physically-motivated LF model to obtain constraints on a possible turn-over of
LF at faint magnitudes. In the empirical model the LF drops fast when the
absolute UV magnitude $M_{\rm UV}$ is much larger than a turn-over absolute UV
magnitude $M_{\rm UV}^{\rm T}$. We obtain $M_{\rm UV}^{\rm T}\gtrsim-14.6 $
(15.2) at 1 (2) $\sigma$ confidence level (C.L.) for $z\sim6$. In the
physically-motivated analytical model, star formation in halos with circular
velocity below $v_c^*$ is fully quenched if these halos are located in ionized
regions. Using updated lensing models and new additional FFs data, we
re-analyze previous constraints on $v_c^*$ and $f_{\rm esc}$ presented by
Castellano et al. 2016a (C16a) using a smaller dataset. We obtain new
constraints on $v_c^*\lesssim 59$ km s$^{-1}$ and $f_{\rm esc}\lesssim 56\%$
(both at 2$\sigma$ C.L.) and conclude that there is no turn-over detected so
far from the analyzed FFs data. Forthcoming {\it JWST} observations will be key
to tight these constraints further.
