Study of minimizers for problems with linear growth under bounded slope condition

Abstract This paper addresses the uniqueness and Lipschitz regularity of solutions for a general minimization problem. We assume that the boundary data satisfy the bounded slope condition to ensure the existence of a minimizer, which is globally Lipschitz continuous. With this regularity, we can prove the existence of a unique solution among functions of bounded variation. Finally, we apply these results to a problem related to Hencky plasticity to prove the continuity of the stress and to investigate the regularity and geometry of the level sets of the minimizer.