the anisotropy be given by the function $\phi(x,Du) = \| Du \|_p$, where $1 < p < \infty$. Let $E$ be a $\phi-$minimal set with respect to $\Omega$, i.e. $\chi_E$ is a function of $\phi-$least gradient in $\Omega$. Then every connected component of $\partial E$ is a line segment.

Preliminaries
=============

Least gradient functions
------------------------

Now we shall briefly recall basic facts about least gradient functions. What we need most in this paper is the Miranda stability theorem and the relationship between functions of least gradient and minimal surfaces. For more information, see [@Giu].

We say that $u \in BV(\Omega)$ is a function of least gradient, if for every compactly supported $($equivalently: with trace zero$)$ $v \in BV(\Omega)$ we have

$$\int_\Omega |Du| \leq \int_\Omega |D(u + v)|.$$

We say that $u \in BV(\Omega)$ is a solution of the least gradient problem in the sense of traces $($solution of LGP$)$ for given $f \in L^1(\Omega)$, if $Tu = f$ and for every $v \in BV(\Omega)$ such that $Tv = 0$ we have

$$\int_\Omega |Du| \leq \int_\Omega |D(u + v)|.$$

To underline the difference between the two notions, we recall a stability theorem by Miranda:

$($[@Mir Theorem 3]$)$ Let $\Omega \subset \mathbb{R}^N$ 