$C^1$-approximation and extension of subharmonic functions

Criteria for the uniform approximability in $\mathbb R^N$, $N\geqslant 2$, of the gradients of $C^1$-subharmonic functions by the gradients of similar functions that are harmonic in neighbourhoods of a fixed compact set are obtained. The semiadditivity of the capacity related to the problem is proved and several metric conditions for the approximation are found. An estimate of the flux of the gradient of a subharmonic function in terms of the capacity of its “sources” and a theorem on the possibility of a $C^1$-extension of a subharmonic function in a ball to a subharmonic function on the whole of $\mathbb R^N$ are established.