Least gradient problems with Neumann boundary condition

We study existence of minimizers of the least gradient problem inf (v is an element of BVg) integral(Omega) phi(x,Du), where BVg = {v is an element of BV(Omega) : integral(partial derivative Omega) gv = 1}, phi(x, p) : Omega x R-n -> R is a convex, continuous, and homogeneous function of degree 1 with respect to the p variable, and g satisfies the compatibility condition integral(partial derivative Omega) gds = 0. We prove that for every 0 not equivalent to g is an element of L-infinity (partial derivative Omega) there are infinitely many minimizers in BV (Omega). Moreover there exists a divergence free vector field T is an element of (L-infinity (Omega))(n) that determines the structure of level sets of all minimizers, i.e. T determines Du/vertical bar Du vertical bar-a.e. in Omega, for every minimizer u. We also prove some existence results for general 1-Laplacian type equations with Neumann boundary condition. A numerical algorithm is presented that simultaneously finds T and a minimizer of the above least gradient problem. Applications of the results in conductivity imaging are discussed. (C) 2017 Elsevier Inc. All rights reserved.