Existence and structure of minimizers of least gradient problems

We study existence of minimizers of the general least gradient problem inf(u is an element of BVf) integral(Omega) phi(x, Du), where BVf = {u is an element of BV(Omega) : u|(partial derivative Omega) = f}, f is an element of L-1(partial derivative Omega), and phi(x, xi) is a convex, continuous, and homogeneous function of degree 1 with respect to the xi variable. It is proven that 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 (possible) minimizers; that is, T determines Du/|Du|, |Du|-almost everywhere in Omega, for all minimizers u. We also prove that every minimizer of the above least gradient problem is also a minimizer of inf u. inf(u is an element of Af) integral(n)(R) phi(x, Du), where A(f) = {v is an element of BV(R-n) : v = f on Omega(c)} and f is an element of W-1,W-1(R-n) is a compactly supported extension of f is an element of L-1(partial derivative Omega), and show that T also determines the structure of level sets of all minimizers of the latter problem. This relationship between minimizers of the above two least gradient problems could be exploited to obtain information about existence and structure of minimizers of the former problem from those of the latter, which always exist.