The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature
The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature
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DOI：
10.1090/s0002-9947-98-02079-0
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发表时间：
1998
影响因子：
1.3
通讯作者：
Bo Guan
中科院分区：
文献类型：
--
作者：
Bo Guan
本文将严格凸域上Monge-Ampere方程Dirichlet问题解的存在性和正则性的著名结果推广到R'上的任意光滑有界域以及一般黎曼流形上。对于非退化情况，我们证明了经典可解的充分必要条件是子解的存在性。对于完全退化的情况，我们证明了当给定的边界数据扩展到局部严格凸c2函数Q时，解在C11(Q)中。作为应用，我们证明了Minkowski空间中跨越规定边界的常高斯-克罗内克曲率的类空间超曲面的存在性。
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 bounded domain in R' as well as in a general Riemannian manifold. We prove for the nondegenerate case that a sufficient (and necessary) condition for the classical solvability is the existence of a subsolution. For the totally degenerate case we show that the solution is in C11(Q) if the given boundary data extends to a locally strictly convex c2 function ori Q. As an application we prove some existence results for spacelike hypersurfaces of constant Gauss-Kronecker curvature in Minkowski space spanning a prescribed boundary.