Approximation of functions on a compact set by solutions of elliptic equations. Quantitative results

TL;DR

This paper demonstrates that functions with generalized Hölder continuity on certain regular compact sets can be approximated by solutions to elliptic equations, with the approximation rate linked to the function's continuity modulus.

Contribution

It provides a quantitative approximation method for Hölder continuous functions on Ahlfors regular sets using elliptic equation solutions, establishing explicit rates of convergence.

Findings

Approximation rate depends on the function's continuity modulus.

Applicable to functions on (m-2)-Ahlfors regular compact sets.

Extends approximation theory with elliptic PDE solutions.

Abstract

We establish that a generalized H\"{o}lder continuous function on an $(m-2)$-Ahlfors regular compact set in $\mathbb{R}^m$ can be approximated by solutions of an elliptic equation, with the rate of approximation determined by the continuity modulus of the function.