Higher order H{\"o}lder approximation by solutions of second order elliptic equations

For a given second order elliptic operation $\mathcal{L}$ in a domain $\Omega\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\$, and a compact set $\mathbf{K}\subset\Omega$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space $\mathcal{H}^{\mathbf{r}+\omega}_{\mathcal{L}}(\mathbf{K})$ 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.