Regularity of Solutions

We now present several regularity results concerning solutions to the (anisotropic) least gradient problem. Due to the linear growth of the corresponding functional, the scope of the regularity results is quite limited even in the isotropic case, and even for smooth boundary data it is possible that the solution is only Lipschitz. Furthermore, as we saw in the previous Chapter, in the anisotropic case uniqueness of solutions for continuous boundary data requires strong assumptions on the metric integrand, and if they are violated then also regularity of solutions breaks down. Therefore, we will focus on the description of several phenomena: continuity and Hölder continuity of solutions in some restricted settings; higher integrability of solutions in general; and local boundedness of solutions.