  We study existence of minimisers to the least gradient problem on a strictly
convex domain in two settings. On a bounded domain, we allow the boundary data
to be discontinuous and prove existence of minimisers in terms of the Hausdorff
measure of the discontinuity set. Later, we allow the domain to be unbounded,
prove existence of minimisers and study their properties in terms of the
regularity of boundary data and the shape of the domain.
