Existence and uniqueness of minimizers of general least gradient problems

TL;DR

This paper establishes existence, uniqueness, and comparison results for minimizers of general least gradient problems, with sharp regularity conditions on the weight function, motivated by conductivity imaging applications.

Contribution

It proves sharp conditions for existence and uniqueness of minimizers in general least gradient problems, including weighted cases, and constructs counterexamples for regularity assumptions.

Findings

Existence and uniqueness of minimizers under certain conditions.

Counterexamples showing sharpness of regularity assumptions.

Uniqueness in weighted least gradient problems with $a ot= 0$.

Abstract

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems - under certain sharp conditions - for minimizers of the general least gradient problem \[\inf_{u\in BV_f(\Omega)} \int_{\Omega}\varphi(x,Du),\] where $f:\partial \Omega\to \R$ is continuous, \[ BV_f(\Omega):=\{v\in BV(\Omega): \ \ \forall x\in \partial \Omega, \ \ \lim_{r\to 0} \ \esssup_{y\in \Omega, |x-y|<r} |f(x) - v(y)| = 0 \ \} %BV_f(\Omega)=\{u\in BV(\Omega): {0.1cm} u|_{\partial \Omega}=f {0.1cm} \hbox{and} {0.1cm} {0.1cm} u {0.1cm} \hbox{is continuous at} {0.1cm} \partial \Omega \}. \] and $\varphi(x,\xi)$ is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the $\xi$ variable. In particular we prove that if $a\in C^{1,1}(\Omega)$ is bounded away from zero, then minimizers of the weighted least gradient problem $\inf_{u \in BV_f}\int_{\Omega} a|Du|$ are unique in $BV_f(\Omega)$. We construct counterexamples to show that the regularity assumption $a\in C^{1,1}$ is sharp, in the sense that it can not be replaced by $a\in C^{1,\alpha}(\Omega)$ with any $\alpha<1$.