Some minimization problems in the class of convex functions with prescribed determinant

Abstract

We consider minimizers of linear functionals of the type

$$L\left(u\right)={\int }_{\partial \Omega }ud\sigma -{\int }_{\Omega }udx$$

in the class of convex functions $u$ with prescribed determinant $det{D}^{2}u=f$.

We obtain compactness properties for such minimizers and discuss their regularity in two dimensions.