Stable minimal cones with an isolated singularity

The thesis is divided in two chapters. In the first chapter we prove the Simons' theorem, about the non existence, in dimension n<8, of cones with an isolated singularity that are minimizers for the perimeter. This, together with the Federer's dimension reduction principle and the epsilon-regularity of the reduced boundary, allow us to estabilish that, in dimension n<8, all the perimeter minimizers are smooth. In the second chapter we treat a generalization of this problem, following a work of Edelen, Chodosh, and Li. They work on sets of locally finite perimeter cutted by an half space, studying the minima of a functional whose stationary points model the equilibrium state of incompressible fluid. This functional is dependent on an angle, that turns out to be the fixed contact angle that a smooth stationary point forms with the contact hyperplane. Also in this case we can reduce ourselves to the study of cones with an isolated singularity, and we get results of regularity depending on the dimension and on the contact angle. In the case where the ambient space has dimension n+1=4, we give a new proof of the same result of Chodosh, Edelen, and Li, adding an assumption.