• Corpus ID: 37616041

The Monge-Ampµere equation and its geometric applications

@inproceedings{Trudinger2008TheME,
  title={The Monge-Ampµere equation and its geometric applications},
  author={Neil S. Trudinger and Xu-jia Wang},
  year={2008}
}
In this paper we present the basic theory of the Monge-Ampµere equation together with a selection of geometric applications, mainly to a‐ne geometry. First we introduce the Monge-Ampµere measure and the resultant notion of generalized solution of Aleksandrov. Then we discuss a priori estimates and regularity, followed by the existence and uniqueness of solutions to various boundary value problems. As applications we consider the existence of smooth convex hypersurfaces of prescribed Gauss… 
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References

SHOWING 1-10 OF 209 REFERENCES
The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature
In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampere equations in a strictly convex domain to an arbitrary smooth
Boundary regularity for the Monge-Ampere and affine maximal surface equations
In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomoge- neous term is only assumed to be Holder continuous.
Sur les equations de Monge-Ampère
In this paper we study the real Monge-Arapere equations: det(D2u)= f(x) in 0, u convex in 0, u=0 on ∂0, and we introduce a new method for solving these equations which enables us to show the
Recent developments in elliptic partial differential equations of Monge-Ampere type
In conjunction with applications to optimal transportation and conformal geometry, there has been considerable research activity in recent years devoted to fully nonlinear, elliptic second order
EXISTENCE OF CONVEX HYPERSURFACES WITH PRESCRIBED GAUSS-KRONECKER CURVATURE
Let f(x) be a given positive function in Rn+1. In this paper we consider the existence of convex, closed hypersurfaces X so that its GaussKronecker curvature at x ∈ X is equal to f(x). This problem
On the second boundary value problem for Monge-Ampère type equations and optimal transportation
This paper is concerned with the existence of globally smooth so- lutions for the second boundary value problem for certain Monge-Amp` ere type equations and the application to regularity of
REAL ANALYSIS RELATED TO THE MONGE-AMPERE EQUATION
In this paper we consider a family of convex sets in Rn, F = {S(x, t)}, x ∈ Rn, t > 0, satisfying certain axioms of affine invariance, and a Borel measure μ satisfying a doubling condition with
Dirichlet problems for general Monge-Ampere equations
Here 172z denotes the Hessian of z with respect to a given metric d s2= gu d u~d u a, in the local chart, VZz =z u F~} z~ where (~ are the Christoffel symbols. There have been many papers to devoted
BOUNDEDLY NONHOMOGENEOUS ELLIPTIC AND PARABOLIC EQUATIONS IN A DOMAIN
In this paper the Dirichlet problem is studied for equations of the form and also the first boundary value problem for equations of the form , where and are positive homogeneous functions of the
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