Existence and uniqueness of minimizers of general least gradient problems

Abstract

AbstractMotivated 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), wheref:\partial\Omega\to\mathbb{R}is continuous,BV_{f}(\Omega):=\bigl{\{}v\in BV(\Omega):\lim_{r\to 0}\operatornamewithlimits{% ess\,sup}_{y\in\Omega,|x-y|<r}|f(x)-v(y)|=0\text{ for }x\in\partial\Omega\bigr% {\}}and\varphi(x,\xi)is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular, we prove that ifa\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 inBV_{f}(\Omega). We construct counterexamples to show that the regularity assumptiona\in C^{1,1}is sharp, in the sense that it can not be replaced bya\in C^{1,\alpha}(\Omega)with any\alpha<1.