Removability of H\"older graphs for continuous sobolev functions

TL;DR

This paper characterizes when H"older-$\alpha$ graphs can be removed for continuous Sobolev $W^{1,2}$ functions, showing a threshold at $\alpha=2/3$ for removability.

Contribution

It provides a precise characterization of the removability of H"older-$\alpha$ graphs for continuous Sobolev functions, establishing a critical exponent at $\alpha=2/3$.

Findings

Graphs with $\alpha > 2/3$ are removable.

Graphs with $\alpha < 2/3$ are not necessarily removable.

The paper identifies a sharp threshold at $\alpha=2/3$.

Abstract

We characterize the removability of H\"older-$\alpha$ graphs with respect to continuous Sobolev $W^{1,2}$ functions. For $\alpha > 2/3$ these graphs are removable, while for $\alpha <2/3$ there exist graphs which are not removable.