Let
M
\mathcal {M}
be a second order, linear, parabolic partial differential operator with coefficients defined in a domain
D
=
Ω
×
(
0
,
T
)
\mathcal {D} = \Omega \times (0,\,T)
in
R
n
×
R
{{\mathbf {R}}^n} \times {\mathbf {R}}
, with
Ω
\Omega
a domain in
R
n
{{\mathbf {R}}^n}
. Let
u
u
be a suitably regular real function in
D
\mathcal {D}
such that
u
u
is bounded below and
M
u
\mathcal {M}u
is bounded above in
D
\mathcal {D}
. If
u
⩾
0
u \geqslant 0
on
Ω
×
{
0
}
\Omega \times \{ 0\}
except on a set
Γ
×
{
0
}
\Gamma \times \{ 0\}
, with
Γ
\Gamma
a subset of
Ω
\Omega
of suitably restricted Hausdorff dimension, then necessarily
u
⩾
0
u \geqslant 0
also on
Γ
×
{
0
}
\Gamma \times \{ 0\}
. The allowable Hausdorff dimension of
Γ
\Gamma
depends on the coefficients of
M
\mathcal {M}
. For example, if
M
\mathcal {M}
is the heat operator
Δ
−
∂
/
∂
t
\Delta - \partial /\partial t
, the Hausdorff dimension of
Γ
\Gamma
needs to be smaller than the number of space dimensions
n
n
. Analogous results are valid for exceptional boundary sets on the lateral boundary,
∂
Ω
×
(
0
,
T
)
\partial \Omega \times (0,\,T)
, of
D
\mathcal {D}
.