Characterization and semiadditivity of the $\mathcal C^1$-harmonic capacity

The $\mathcal C^1$-harmonic capacity $\kappa^c$ plays a central role in problems of approximation by harmonic functions in the $\mathcal{C}^1$-norm in $\mathbb{R}^{n+1}$. In this paper we prove the comparability between the capacity $\kappa^c$ and its positive version $\kappa^c_+$. As a corollary, we deduce the semiadditivity of $\kappa^c$. This capacity can be considered as a generalization in $\mathbb{R}^{n+1}$ of the continuous analytic capacity $\alpha$ in $\mathbb{C}$. Moreover, we also show that the so-called inner boundary conjecture fails for dimensions $n>1$, unlike in the case $n=1$.