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The affine Plateau problem

Authors: Neil S. Trudinger and Xu-Jia Wang
Journal: J. Amer. Math. Soc. 18 (2005), 253-289
MSC (2000): Primary 35J40; Secondary 53A15
Published electronically: January 3, 2005
MathSciNet review: 2137978
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Abstract: In this paper, 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.

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Additional Information

Neil S. Trudinger
Affiliation: Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200, Australia

Xu-Jia Wang
Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia

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-Amp\`{e}re measure, curvature measure, Gauss mapping, locally convex hypersurface, generalized Legendre transform
Received by editor(s): September 3, 2003
Published electronically: January 3, 2005
Additional Notes: This research was supported by the Australian Research Council
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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