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The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature


Author: Bo Guan
Journal: Trans. Amer. Math. Soc. 350 (1998), 4955-4971
MSC (1991): Primary 35J65, 35J70; Secondary 58G20
DOI: https://doi.org/10.1090/S0002-9947-98-02079-0
MathSciNet review: 1451602
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Abstract: In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampère equations in a strictly convex domain to an arbitrary smooth bounded domain in $\mathbb R^n$ 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 $C^{1,1} (\overline {\Omega})$ if the given boundary data extends to a locally strictly convex $C^2$ function on $\overline {\Omega}$. As an application we prove some existence results for spacelike hypersurfaces of constant Gauss-Kronecker curvature in Minkowski space spanning a prescribed boundary.


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Additional Information

Bo Guan
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: guan@math.utk.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02079-0
Received by editor(s): August 11, 1995
Received by editor(s) in revised form: November 11, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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