The affine Plateau problem
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Authors
Trudinger, Neil S.
Wang, Xu-Jia
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American Mathematical Society
Abstract
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.
Description
Keywords
Affine Plateau problem, affine maximal hypersurface, affine area functional, affine maximal surface equation, variational problem, second boundary value problem, a priori estimates, strict convexity, interior regularity, Bernstein Theorem, Monge-Ampère measure, curvature measure, Gauss mapping, locally convex hypersurface, generalized Legendre transform
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Journal of the American Mathematical Society