Let
Ω
\Omega
be an open set in
R
2
{{\mathbf {R}}^2}
which is locally convex at each point of its boundary except one, say
(
0
,
0
)
(0,0)
. Under certain mild assumptions, the solution of a prescribed mean curvature equation on
Ω
\Omega
behaves as follows: All radial limits of the solution from directions in
Ω
\Omega
exist at
(
0
,
0
)
(0,0)
, these limits are not identical, and the limits from a certain half-space
(
H
)
(H)
are identical. In particular, the restriction of the solution to
Ω
∩
H
\Omega \cap H
is the solution of an appropriate Dirichlet problem.