Characteristic functions on the boundary of a planar domain need not be traces of least gradient functions

TL;DR

This paper demonstrates that in a smooth bounded planar domain, there exist boundary characteristic functions that are not traces of least gradient functions, extending previous results from the disc to more general domains.

Contribution

The authors generalize the construction of characteristic functions not arising as traces of least gradient functions from the disc to any smooth bounded planar domain.

Findings

Existence of boundary characteristic functions not traceable as least gradient functions in smooth domains

Extension of previous disc-specific results to general smooth planar domains

Provides a new understanding of boundary behavior in least gradient problems

Abstract

Given a smooth bounded planar domain, we construct a compact set on the boundary s.t. its characteristic function is not the trace of a least gradient function. This generalize the construction of Spradlin and Tamasan [ST14] on the disc.