Least Gradient Problem and Minimal Surfaces

In this Chapter, we present the classical results concerning functions of locally least gradient (see Definition 1.1 below), starting with the works of Miranda [126] and Bombieri-De Giorgi-Giusti [19] in the late sixties. We give the proofs of the main results in these works, namely a stability property of least gradient functions and the fact that their superlevel sets are area-minimizing. Then, we restrict our attention to bounded Lipschitz domains in \(\mathbb {R}^N\) , formulate our main definition of least gradient functions (Definition 1.6) and of the least gradient problem, and discuss its relationship to other definitions appearing in the literature. Finally, we present a construction due to Sternberg-Williams-Ziemer [168], which yields existence of a unique and continuous solution to the least gradient problem for continuous boundary data, under geometric assumptions on the domain slightly weaker than strict convexity.