Existence and uniqueness of minimizers of general least gradient problems

Robert L. Jerrard; Amir Moradifam; Adrian I. Nachman

Paper

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

where

is continuous,

and

is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the

variable. In particular we prove that if

is bounded away from zero, then minimizers of the weighted least gradient problem

are unique in

. We construct counterexamples to show that the regularity assumption

is sharp, in the sense that it can not be replaced by

with any