Higher order H{ö}lder approximation by solutions of second order elliptic equations

For a given second order elliptic operation $\mathcal{L}$ in a domain $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathbf{N}}$, \mathbf{N}\, and a compact set $\mathbf{K}\subset \mathrm{\Omega }$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space ${\mathcal{H}}_{\mathcal{L}}^{\mathbf{r}+\omega }(\mathbf{K})$ of continuous functions $f(x),x\in \mathbf{K}$, admitting, for any $\delta >0$, a local approximation in the $\delta$-neighborhood of any point $x\in \mathbf{K}$, with ${\delta }^{\mathbf{r}}\omega (\delta )$-error estimate, by solutions of the equation $\mathcal{L}u=0$. For such functions, we prove the existence of a global approximation $v_{\delta }$ on $\mathbf{K}$ with the same order of error estimate, by a solution of the same equation in a $\delta$-neighborhood of $\mathbf{K}$. A number of properties of these functions $v_{\delta }$ and their derivatives are established.