*preprint*

**Inserted:** 9 may 2023

**Year:** 2023

**Abstract:**

We deal with suitable nonlinear versions of Jauregui's Isocapacitary mass in $3$-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1<p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's Isoperimetric mass, as $p \to 1^+$. We derive Positive Mass Theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the $\mathrm{ADM}$ mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose Inequality in the optimal asymptotic regime.