| In this thesis two problems are considered. The Monge-Kantorovich theory appears as a material tool.; In the first problem we consider the fully compressible semi-geostrophic system, an approximate model in meteorology, which is used to model large-scale behaviour of the atmosphere, in particular, the formation of atmospheric fronts. We derive a reformulation in so-called geostrophic coordinates, for which we show existence of stable weak solutions through a time-discretization. We also answer several open questions and confirm several predictions made by meteorologists. Furthermore we obtain time regularity of the solutions both in physical and geostrophic variables. The Monge-Kantorovich theory appears because of the stability requirement; we extend the theory to more general costs.; In the second problem we show existence and uniqueness of minimizers for a class of polyconvex functionals. The direct methods of the calculus of variations fail due to a lack of coercivity. A relaxation and corresponding dual problem are introduced. An analysis of the geometry of a minimizer in the relaxation leads to an equivalent problem, which is the essential step. It establishes a connection with the Monge-Kantorovich theory. The Euler-Lagrange equations for the new problem are central and establish duality and existence and uniqueness of a minimizer for the primal problem. They also provide sufficiency conditions. |
