Electron. J. Diff. Eqns., Vol. 2006(2006), No. 138, pp. 1-9.

The Dirichlet problem for the Monge-Ampere equation in convex (but not strictly convex) domains

David Hartenstine

Abstract:
It is well-known that the Dirichlet problem for the Monge-Ampere equation $\det D^2 u = \mu$ in a bounded strictly convex domain $\Omega$ in $\mathbb{R}^n$ has a weak solution (in the sense of Aleksandrov) for any finite Borel measure $\mu$ on \Omega and for any continuous boundary data. We consider the Dirichlet problem when \Omega is only assumed to be convex, and give a necessary and sufficient condition on the boundary data for solvability.

Submitted April 29, 2006. Published October 31, 2006.
Math Subject Classifications: 35J65, 35D05.
Key Words: Aleksandrov solutions; Perron method; viscosity solutions.

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David Hartenstine
Department of Mathematics
Western Washington University
516 High Street, Bond Hall 202
Bellingham, WA 98225--9063, USA
email: david.hartenstine@wwu.edu

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