The affine Plateau problem. (English) Zbl 1229.53049
Summary: We study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 53A15 | Affine differential geometry |
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