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We show an improvement of Bray sharp mass-capacity inequality and Bray-Miaosharp upper bound of the capacity of the boundary in terms of its area, forthree-dimensional, complete, one-ended asymptotically flat manifolds withcompact, connected boundary and with nonnegative scalar curvature, underappropriate assumptions on the topology and on the mean curvature of theboundary. Our arguments relies on two monotonicity formulas holding along levelsets of a suitable harmonic potential, associated to the boundary of themanifold. This work is an expansion of the results contained in the PhD thesisof the author.

ADM mass, area and capacity in asymptotically flat $3$-manifolds with nonnegative scalar curvature