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

Abstract:

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 $n=1$.