Nonstandard Boundary Conditions

In this Chapter, we study three variants of the least gradient problem related to the shape of the domain. The first one concerns the case when we impose a Dirichlet boundary condition only on a part of the boundary; in Chap. , we studied this problem for weak solutions, and now we impose the boundary condition in the trace sense. The second variant concerns the case when the domain is a rectangle in two dimensions. This is an example of the domain which is convex, but not strictly convex, therefore most of the methods used in previous Chapters do not work; for instance, we cannot directly apply the Sternberg-Williams-Ziemer construction to obtain existence of minimizers. Instead, we solve the least gradient problem using a sequence of approximate problems on strictly convex domains. The third variant concerns the situation when the domain is unbounded, and in particular how it affects uniqueness of solutions. We highlight the issues that are different from the standard least gradient problem. For clarity, in this Chapter we will present the reasoning in the isotropic setting.