52 2. FOUR IMPORTANT LINEAR PDE
The region UT
DEFINITIONS.
(i) We define the parabolic cylinder
UT := U × (0,T ].
(ii) The parabolic boundary of UT is
ΓT :=
¯
U
T
− UT .
We interpret UT as being the parabolic interior of
¯
U × [0,T ]: note care-
fully that UT includes the top U × {t = T }. The parabolic boundary ΓT
comprises the bottom and vertical sides of U × [0,T ], but not the top.
We want next to derive a kind of analogue to the mean-value property for
harmonic functions, as discussed in §2.2.2. There is no such simple formula.
However let us observe that for fixed x the spheres ∂B(x, r) are level sets of
the fundamental solution Φ(x−y) for Laplace’s equation. This suggests that
perhaps for fixed (x, t) the level sets of fundamental solution Φ(x − y, t − s)
for the heat equation may be relevant.
DEFINITION. For fixed x ∈
Rn,
t ∈ R, r 0, we define
E(x, t; r) := (y, s) ∈
Rn+1
| s ≤ t, Φ(x − y, t − s) ≥
1
rn
.
