  Screening of a strongly charged macroion by multivalent counterions is
considered. It is shown that counterions form a strongly correlated liquid at
the surface of the macroion. Cohesive energy of this liquid leads to additional
attraction of counterions to the surface which is absent in conventional
solutions of Poisson-Boltzmann equation. Away from the surface this attraction
can be taken into account by a new boundary condition for the concentration of
counterions near the surface. Poisson-Boltzmann equation is solved with this
boundary condition for a charged flat surface, a cylinder and a sphere. In all
three cases, screening is much stronger than in the conventional approach. At
some critical exponentially small concentration of multivalent counterions in
the solution they totally neutralize the surface charge at small distances from
the surface. At larger concentrations they invert the sign the net macroion
charge. Absolute value of the inverted charge density can be as large as 20% of
that of the bare one. In particular, for a cylindrical macroion, it is shown
that for screening by multivalent counterions predictions of the
Onsager-Manning theory are quantitatively incorrect. The net charge density of
the cylinder is smaller than their theory predicts and inverts sign with
growing concentration of counterions. Moreover the condensation looses its
universality and the net charge linear density depends on the bare one.
