Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 138, pp. 1-9.
Title: The Dirichlet problem for the Monge-Ampere
equation in convex (but not strictly convex) domains
Author: David Hartenstine (Western Washington Univ., Bellingham, WA, USA)
Abstract:
It is well-known that the Dirichlet problem for the Monge-Amp\`ere
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.