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

.