The affine Plateau problem

By Neil S. Trudinger, Xu-jia Wang. 2004

107 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/a47c1226972bd9271e2f096e3748a969ed3607d6

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In Applied Mathematics

Citations in the corpus

Relative entropy of cone measures and Lp centroid bodies (2009, background): “N. S. Trudinger and X. Wang, ‘The affine Plateau problem’, J. Amer.”
The Centro-Affine Hadwiger Theorem (2013, background): “area (a long conjectured property, finally established in [48]) was crucial for the solution of the affine Plateau problem [72].”
Functional Versions of Lp-Affine Surface Area and Entropy Inequalities (2014, unknown): “The Lp-affine surface area and its related inequalities play a major role in convex and differential geometry [2, 22, 24, 30, 31, 34, 43, 45, 47, 48, 50, 51, 52].”
Boundary regularity for the Monge-Ampere and affine maximal surface equations (2005, background): “(1.5) The second boundary value problem for (1.4) (as introduced in [21]), is the Dirichlet problem for the system (1.4), (1.5), that is to prescribe u = ϕ, w = ψ on ∂Ω.(1.6) We will prove Theorem 1.2.”
A classification of $\operatorname{SL}(n)$ invariant valuations (2010, background): “The long conjectured upper semicontinuity of affine surface area (for smooth surfaces as well as for general convex surfaces) was proved by Lutwak [37] in 1991 and was important in the solution of the affine Plateau problem by Trudinger and Wang [61] in 2005.”
The Spherical Convex Floating Body (2014, unknown): “In particular the upper semicontinuity proved to be critical in the solution of the affine Plateau problem by N. S. Trudinger and X.-J. Wang [77].”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “Another classical instance where the Monge-Ampère equation arises is affine geometry, more precisely in the “affine sphere problem” and the “affine maximal surfaces” problem (see for instance [28, 98, 30, 107, 108, 109]).”
C1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two (2009, methodology): “We choose to present separately the proof for c(x, y) =− x · y: indeed the proof in this case is much shorter, and it might help the reader to follow the argument in the general case (see also for example [8, 9 ], where similar techniques are employed).”
General affine surface areas (2009, unknown): “Numerous important results regarding equi-affine surface area were obtained in recent years (see, for example, [1,2,45,48–51]).”