Variational inequalities connected with linear elliptic systems of general form are considered. The vector-valued functions, satisfying the variational inequalities in the domain of definition, take values in an arbitrary fixed convex set of the space R^N, N>1. It is shown that the second derivatives of the solutions belong to the space L_2,α. An analogous result for a problem with a constraint of special form on the boundary of the domain (the classical Signorini problem) was obtained earlier by Kinderlehrer.