In the first part of the talk we will introduce variational problems involving the free energy associated to the surface tension of liquids.
We will show a first classical way of modelling this energy via De Giorgi's perimeter. Then, we will present a more recent formulation involving the fractional perimeter and we will highlight the main differences between the two settings.
In the second part of the talk we will investigate the problem of studying equilibrium shapes for small liquid drops under the action of a confining bulk potential.
Some of the results that I will present were obtained in collaboration with Fumihiko Onoue (TUM, München) and Matteo Novaga (UNIPI, Pisa).