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

Abstract

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