Uniqueness and Structure of Solutions

This Chapter is devoted to the study of uniqueness and structure of solutions to the anisotropic least gradient problem. As we saw in Chaps. 1 and 2, even in the isotropic case we need to assume continuity of boundary data in order to have a unique solution. In the anisotropic case, it turns out that uniqueness of solutions strongly depends on the regularity of the metric integrand \(\phi \) and the shape of the domain. In this Chapter, we first prove an anisotropic counterpart of the comparison principle given in Theorem 1.19, under the assumption that the metric integrand \(\phi \) is uniformly convex and sufficiently smooth. Then, we show some examples highlighting what may happen for less regular metric integrands. The second half of the Chapter is focused on the isotropic case, where show a decomposition property for functions of least gradient and study in detail the structure of solutions.