Not all traces on the circle come from functions of least gradient in the disk

TL;DR

This paper demonstrates that not all boundary traces on a circle originate from least gradient functions in the disk, highlighting the necessity of continuity assumptions in classical existence and uniqueness theorems.

Contribution

It provides a counterexample showing boundary data continuity cannot be omitted in least gradient problem solutions.

Findings

Counterexample of an $L^1$ boundary trace not from a least gradient function

Continuity of boundary data is essential for existence and uniqueness theorems

Challenges assumptions in classical least gradient problem results

Abstract

We provide an example of an $L^1$ function on the circle, which cannot be the trace of a function of bounded variation of least gradient in the disk. This shows that in theorems on existence and uniqueness of solutions to the least gradient problem, proven by Sternberg, Williams and Ziemer and published in 1992-1993, the hypothesis that the boundary data is continuous cannot be removed.