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 \[\inf_{u\in BV_f(Ω)} \int_Ωφ(x,Du),\] where $f:\partial \Omega \to \mathbb{R}$ is continuous, \[ BV_f(Ω):=\{v\in BV(Ω): \ \ \forall x\in \partial Ω, \ \ \lim_{r\to 0} \ \esssup_{y\in Ω, |x-y|<r} |f(x) - v(y)| = 0 \ \} %BV_f(Ω)=\{u\in BV(Ω): {0.1cm} u|_{\partial Ω}=f {0.1cm} \hbox{and} {0.1cm} {0.1cm} u {0.1cm} \hbox{is continuous at} {0.1cm} \partial Ω\}. \] and $\phi (x,\xi )$ 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 ${\inf }_{u\in B{V}_f}{\int }_{\Omega }a|Du|$ are unique in $B{V}_f(\Omega )$. 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 $\alpha <1$.