Study of some variational problems in Riemannian geometry and mathematical economics.

Summary In the first part of this thesis, we prove a weak version of the following conjecture, formulated in 1983 by Hildebrandt: in dimension 2, applications which are critical points of a conformal diffeomorphism invariant functional are regular. We show the regularity of such applications, provided that they are a priori bounded. This result extends the theorem of F. Helein on the regularity of harmonic applications with values in a compact riemannian variety without edge. The variational problems studied in the second part are motivated by economic questions. They consist in the maximization of some functionals on the cone of convex functions. We give a sufficient condition for the convexity constraint to be active. This condition, which involves in particular the geometry of the domain, is verified in very common situations in economics. Typically in dimension 2, there exists a region where the rank of the hessian of the solution is 1. We write the Euler equations of the problem using so-called "sweep" operators. We explain how to use the sweep conditions to construct the solution of the problem. However, this construction requires some prior knowledge of the form of the solution. This leads us to study the problem of the numerical approximation of the solution: the difficulty lies in determining the directions in which the convexity constraint is saturated. We explore various finite element methods and show that the simplest of them face a serious theoretical obstruction.