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. The affine Plateau problem may be formulated as follows. LetM0 be a bounded, connected hypersurface in Euclidean (n + 1)-space, R, with smooth boundary Γ = ∂M0. Assume that M0 ∪ Γ is smooth and locally uniformly convex up to boundary. Let S[M0] denote the set of locally uniformly convex hypersurfaces M with boundary Γ, which can be smoothly deformed fromM0 in the family of locally uniformly convex hypersurfaces whose Gauss mapping images lie in that of M0. A hypersurface M ⊂ R is called locally uniformly convex if it is a C immersion of an n-manifold N , whose principal curvatures are everywhere positive, and the Gauss mapping of M is the mapping G : M → S which assigns to every point in M its unit normal vector. The affine metric (also called the Berwald-Blaschke metric) on a locally uniformly convex hypersurface M is defined by