Monotone Quantities for $p$-Harmonic functions and the Sharp $p$-Penrose inequality

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $\mathrm{\Sigma }$. Fix a number $1<p<3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $\mathrm{\Sigma }$ in $M$, which was first proved by Xiao using weak inverse mean curvature flow.