  In this paper we establish the $C^{1,\beta}$ regularity of the regular part
of the free boundary in the Signorini problem for elliptic operators with
variable Lipschitz coefficients. This work is a continuation of the recent
paper [GSVG14], where two of us established the interior optimal regularity of
the solution. Two of the central results of the present work are a new
monotonicity formula and a new epiperimetric inequality.
