We provide an analysis of the least gradient problem in the case when the boundary datum is only imposed on a part of the boundary. First, we give a characterisation of solutions in a general setting using convex duality theory. Then, we discuss the way in which solutions attain their boundary values, structure of solutions and their regularity.

中文翻译：

---

部分边界上狄利克雷条件的最小梯度问题

我们分析了边界基准仅施加在边界的一部分上的情况下的最小梯度问题。首先，我们使用凸对偶理论给出一般情况下解的表征。然后，我们讨论解获得边界值的方式、解的结构及其规律性。