The trace space of anisotropic least gradient functions depends on the anisotropy

TL;DR

This paper investigates how the trace space of anisotropic least gradient functions varies with different anisotropic norms, demonstrating that the trace space depends on the specific anisotropy and providing explicit examples.

Contribution

It establishes that the trace space of anisotropic least gradient functions depends on the anisotropic norm and characterizes when trace spaces coincide based on the norms.

Findings

Trace spaces depend on the anisotropic norm.

Coincidence of trace spaces occurs only when norms coincide.

Explicit example of a characteristic function in one trace space but not the other.

Abstract

We study the set of possible traces of anisotropic least gradient functions. We show that even on the unit disk it changes with the anisotropic norm: for two sufficiently regular strictly convex norms the trace spaces coincide if and only if the norms coincide. The example of a function in exactly one of the trace spaces is given by a characteristic function of a suitably chosen Cantor set.