Geometric variational problems in the setting of sets of finite perimeter

The theory of sets of finite perimeter provides, in the broader framework of Geometric Measure Theory, a particularly wellsuited framework for studying the existence, regularity, and structure of singularities of minimizers in those geometric variational problems in which surface area is minimized under a volume constraint. To this end, the class of sets of finite perimeter satisfy these requirements: 1. a class F of sets E ⊂ Rn endowed with a topology with good compactness properties so that sets with smooth boundaries belong to this class and are dense; 2. a notion of perimeter P (E) for every E ∈ F so that E → P (E) is lower- semicontinuous on F, and P extends the usual notion of perimeter Hn−1 of the boundary ∂E; more precisely we require that P (E) = Hn−1(∂E) for every set E with smooth boundary and for every E ∈ F there exists a sequence Eh → E with smooth boundaries and satisfying Hn−1(∂Eh) → P (E). The methods and ideas introduced are applied to study the classical Plateau problem, which consists in finding surfaces with minimal area and prescribed boundary, and variational problems concerning confined liquid drops, briefly denoted as capillarity problems. The equilibrium shape of the liquid drop is given by the non-trivial interaction between the surface tension, which depends on the perimeter of the free surface of the drop inside the container, the contact surface between the drop and the container and the potential energy acting on the drop, for instance gravity. A typical problem is the following: find the domain D ⊂ R3 which minimizes: Area(∂D) + additional integral term + additional constraint (e.g. Volume(D) is prescribed) Thus, as usually done in the calculus of variations, the semicontinuity and compact- ness method is used for proving the existence of minimizers. Geometric properties of the minimizers are deduced by performing first variations of minimizers; for instance, Young’s law comes out naturally as a stationarity condition of the minimizers in the capillarity problems