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