The affine Plateau problem

In this paper we study the Plateau problem for affine maximal hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. In particular we formulate the affine Plateau problem as a geometric variational problem for the affine area functional, and we prove the existence and regularity of maximizers. As a special case, we obtain corresponding existence and regularity results for the variational Dirichlet problem for the fourth order affine maximal surface equation, together with a uniqueness result for generalized solutions