The present thesis is divided into three parts. In the first part, we analyze a
suitable regularization \u2014 which we call nonlinear multidomain model \u2014 of the
motion of a hypersurface under smooth anisotropic mean curvature flow. The
second part of the thesis deals with crystalline mean curvature of facets of a
solid set of R^3 . Finally, in the third part we study a phase-transition model for
Plateau\u2019s type problems based on the theory of coverings and of BV functions
