The Monge-Ampµere equation and its geometric applications

By Neil S. Trudinger, Xu-jia Wang. 2008

274 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/3e98fded69b4c929d882393a609831df67a48e92

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The Monge–Ampère equation and its link to optimal transportation (2013, background): “have found applications in several other problems, such as isometric embeddings, reflector shape design, and in the study of special Lagrangian sub-manifolds, prescribing Weingarten curvature, and in complex geometry on toric manifolds (see for instance the survey paper [110] for more details on the”
On the Dirichlet problem for Monge-Ampère type equations (2013, background): “These equations have attracted significant interest in recent years; for a recent survey and the earlier history see, for example, [26].”
Boundary behavior of large solutions to the Monge-Ampère equations with weights (2015, background): “For the existence, regularity theory and other properties of solutions to the Monge–Ampère equations, see, for instance, [3,6,7,9,17,19,21,29–31,40,42] and the references therein.”
Monge–Ampère equation on exterior domains (2013, background): “The readers may see the survey paper of Trudinger and Wang [37] for more description and applications.”
Rotationally symmetric solutions to the L p -Minkowski problem (2013, background): “It is known that for any non-degenerate convex body K , there is a unique ellipsoid E which attains the minimum volume among all ellipsoids containing K [19].”
Finite element approximations of the three dimensional Monge-Ampère equation (2012, background): “Introduction Let u be a smooth solution to the Monge-Ampère equation [7, 20, 27]: det(D(2)u) = f in Ω, (1.”
Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems (2016, unknown): “The study of Minkowski problems has a long history and strong inﬂuence on both the Brunn-Minkowski theory and fully nonlinear partial diﬀerential equations, see [79] and [73].”