I will present the recent tools I have developed to prove existence and regularity properties of the critical points of anisotropic functionals. In particular, I will provide the anisotropic extension of Allard's celebrated rectifiability theorem and its applications to the anisotropic Plateau problem. Three corollaries are the solutions to the formulations of the Plateau problem introduced by Reifenberg, by Harrison-Pugh and by Almgren-David. Furthermore, I will present the anisotropic counterpart of Allard's compactness theorem for integral varifolds. To conclude, I will focus on the anisotropic isoperimetric problem: I will provide the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points among finite perimeter sets. Moreover I will derive stability inequalities associated to this rigidity theorem.
Some of the presented results are joint works with De Lellis, De Philippis, Ghiraldin, Gioffré, Kolasinski and Santilli.