9 sep 2021 -- 10:20 [open in google calendar]

**Abstract.**

Finding graphs of prescribed mean curvature is one of the classical problems of Calculus of Variations. This problem is closely connected with non-parametric minimal surfaces, and it is strongly motivated by capillarity theory. After an introduction of the classical formulation together with some milestone results, we will describe some recent progress obtained in two different directions. First, the relaxation of the regularity assumptions on the domain, leading to the "weak-regularity" hypothesis. Second, the prescribed mean curvature measure problem.