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

TL;DR

This paper introduces a new function space for solutions of second order elliptic equations, enabling higher order Hölder approximation with controlled error, and establishes the existence of global approximations with similar accuracy.

Contribution

It defines a novel space of functions approximable by elliptic solutions with higher order Hölder regularity and proves the existence of global approximations maintaining the same error order.

Findings

Existence of global approximations with controlled Hölder error

Properties of approximating solutions and their derivatives

Extension of local approximation results to global solutions

Abstract

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.