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Criteria for approximability of functions by solutions of homogeneous secondorder elliptic equations (with constant complex coefficients) in the norms ofthe Whitney $C^1$-spaces on compact sets in $\mathbb R^2$ are obtained in termsof the respective $C^1$-capacities. It is proved that the mentioned$C^1$-capacities are comparable to the classic $C$-analytic capacity, and sohave a proper geometric measure characterization.

On $C^1$-approximability of functions by solutions of second order elliptic equations on plane compact sets and $C$-analytic capacity