Existence and uniqueness of minimizers of general least gradient problems

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 $\alpha<1$.