Existence and structure of minimizers of least gradient problems

TL;DR

This paper proves the existence of minimizers for a general least gradient problem and shows that a divergence-free vector field determines their structure, linking solutions within a bounded domain to those in the entire space.

Contribution

It establishes the existence of a divergence-free vector field that characterizes the structure of all minimizers for the least gradient problem and connects bounded domain minimizers to global minimizers.

Findings

Existence of minimizers for the least gradient problem.

Existence of a divergence-free vector field determining minimizer structure.

Relationship between bounded domain and global minimizers.

Abstract

We study existence of minimizers of the general least gradient problem \[\inf_{u \in BV_f} \int_{\Omega}\varphi(x,Du),\] where $BV_f=\{u \in BV(\Omega): \ \ u|_{\partial \Omega}=f\}$, $f\in L^{1}(\partial \Omega)$, and $\varphi(x,\xi)$ is convex, continuous, and homogeneous function of degree $1$ with respect to the $\xi$ variable. It is proven that there exists a divergence free vector field $T\in (L^{\infty}(\Omega))^n$ that determines the structure of level sets of all (possible) minimizers, i.e. $T$ determines $\frac{Du}{|Du|}$, $|Du|-$ a.e. in $\Omega$, for all minimizers $u$. We also prove that every minimizer of the above least gradient problem is also a minimizer of \[\inf_{u\in \mathcal{A}_f} \int_{\R^n}\varphi(x,Du),\] where $\mathcal{A}_f=\{v\in BV(\R^n): \ \ v=f \ \ \hbox{on}\ \ \Omega^c\}$ and $f\in W^{1,1}(\R^n)$ is a compactly supported extension of $f\in L^1(\partial \Omega)$, and show that $T$ also determines the structure of level sets of all minimizers of the latter problem. This relationship between minimizers of the above two least gradient problems could be exploited to obtain information about existence and structure of minimizers of the former problem from that of the latter, which always exist.