Least Gradient Problem

Abstract

For a given continuous function g defined on the boundary of Ω where Ω is a bounded lipschitz domain in ℝ𝑛satisfying some conditions, we consider proving the existence of a function u in the space of BV(Ω) that is equal to g on the boundary in the trace sense, and the total variation of its distributional derivative evaluated over Ω is minimal among all such functions,in addition to proving uniqueness when u belongs to BV(Ω)∩C(Ω̅).The exposition go deeply in the study of BV theory and sets of finite perimeter.