  We show an improvement of Bray sharp mass-capacity inequality and Bray-Miao
sharp upper bound of the capacity of the boundary in terms of its area, for
three-dimensional, complete, one-ended asymptotically flat manifolds with
compact, connected boundary and with nonnegative scalar curvature, under
appropriate assumptions on the topology and on the mean curvature of the
boundary. Our arguments relies on two monotonicity formulas holding along level
sets of a suitable harmonic potential, associated to the boundary of the
manifold. This work is an expansion of the results contained in the PhD thesis
of the author.
