Criteria for $C^m$-approximability of functions by solutions of homogeneous elliptic equations of second order on compact sets in $\mathbb{R}^N$. All cases

Let $\mathcal{L} = \sum_{n_1,n_2=1}^N c_{n_1n_2}\frac{\partial^2}{\partial x_{n_1} \partial x_{n_2}}\,$ be a homogeneous elliptic operator of second order in $\mathbb{R}^N$ ($N\in \{2, 3, \dots\}$) with constant complex coefficients $c_{n_1n_2}$. A function $f$ is called $\mathcal{L}$-analytic on an open set $D \subset \mathbb{R}^N$, if $f \in C^2(D)$ and $\mathcal{L} f = 0$ in $D$.
In the report it is planned to formulate and discuss criteria for $C^m$-approximability (for all $m \geq 0$) of functions on an arbitrary compact set $X$ in $\mathbb{R}^N$ by functions, which are $\mathcal{L}$-analytic on neighbourhoods of the set $X$.
The mentioned criteria (in the individual form), which are analogous to the well-known capacity criteria of A.G. Vitushkin for uniform holomorphic approximations, are found for all stated $N$, $\mathcal{L}$ and $m<2$ by the author and M.Ya. Mazalov in a number of papers of the last several years. For $m \geq 2$ the corresponding criteria were obtained earlier (in 1980-s) by A. O'Farrell and J. Verdera. We shall also discuss the metrical characteristics of all capacities that are used in the considered criteria.