Least Gradient Problem

Abstract

If f is a given function defined on the boundary of a domain Omega in d-dimensional Euclidean space, the least gradient problem (LGP) asks for the following: among all functions u in the space bounded variation functions in Omega, and having boundary values equal to f, does there exist a function that minimizes the set of all L^1 norms of the gradients of such functions? Furthermore, if such a minimizer exists, what further smooth and minimizing properties does it have? The purpose of this thesis is to study this problem in the two dimensional case, where Omega is strictly convex, and to explore the situation where Omega is only convex. The exposition will present a study of level sets of minimizers, as well as the connection, through the co-area theorem, between the properties of those level sets and the minimizing function.