In the least gradient problem with continuous boundary data it is well known that a minimizer exists and is unique. If we relax this assumption, then in many cases we recover existence of minimizers, but even the simplest examples show that uniqueness fails. In this talk, I will discuss this issue and sketch the proof of a structure result for minimizers in the least gradient problem: two least gradient function with the same trace may only differ on a set, on which they are both locally constant