Existence of minimisers in the least gradient problem for general boundary data

Abstract

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.