Let
d
≥
2
d\geq 2
,
D
=
R
d
×
(
0
,
∞
)
D=\mathbb {R}^{d}\times (0,\infty )
, and suppose
u
u
is harmonic in
D
D
and
C
2
C^{2}
on the closure of
D
D
. If the gradient of
u
u
vanishes continuously on a subset of
∂
D
\partial D
of positive
d
d
-dimensional Lebesgue measure and
u
u
satisfies certain regularity conditions, then
u
u
must be identically constant.