Charleton University Mathematics Journal
Website: http://www.sciencenet.cn/u/zjzhang/
Volume 1, Number 2, November 2010
pp. 11-15
MEAN-VALUE PROPERTY OF THE HEAT EQUATION
ZUJIN ZHANG
A BSTRACT. In this paper, we detailed the proof of the mean-value
theorem for the heat equation, see [1] for example.
Let U ⊂ Rn be open and bounded, and T > 0. We give
1. The parabolic cylinder is the parabolic interior
Definition 1.
of Ū × [0, T ]:
UT ≡ U × (0, T ].
2. The parabolic boundary of UT is
ΓT ≡ ŪT − UT ,
which comprises the bottom and vertical sides of U × [0, T ], but not
the top.
In this parabolic cylinder UT , we want to derive a kind of analogue
to the mean-value property for harmonic function. For this purpose,
we introduce
Definition 2. The heat ball E(x, t; r)(r > 0) at (x, t) ∈ Rn+1 is
)
(
1
n+1
E(x, t; r) = (y, s) ∈ R ; Φ(x − y, t − s) ≥ n .
r
Remark 3.
1. The heat ball is a region in space-time, the boundary of
which is a level set of Φ(x − y, t − s).
Key words and phrases. heat equation, mean-value property, fundamental
solution.
11
12
ZUJIN ZHANG
2. Written explicitly, we have
2
1
1
− |x−y|
4(t−s) = Φ(x − y, t − s) ≥
e
,
rn
[4π(t − s)]n/2
|x−y|2
rn e− 4(t−s) ≥ [4π(t − s)]n/2 .
Applying the logarithmical function, we obtain
n ln r −
n
|x − y|2
≥ ln [4π(t − s)] ,
4(t − s) 2
|x − y|2 ≤ 2n(t − s) ln
r2
.
4π(t − s)
One then verifies easily that RHS of the above inequality equal 0 if
s=t−
r2
or s = t.
4π
This echoes the notion of heat ball, a region in space-time, with the
scale in t is twice that in x.
3. By the above calculations, we find that the function
n
|x − y|2
ψ ≡ − ln [4π(t − s)] −
+ n ln r,
2
4(t − s)
(1)
vanishes on ∂E(x, t; r), which is helpful in integration by parts formula, as we shall in later on. Notice also that
ψy = −
ψs =
y
,
2(t − s)
n s
|x − y|2
−
.
2 t − s 4 (t − s)2
(2)
(3)
Now, we state and prove our mean-value theorem for the heat
equation as
MEAN-VALUE PROPERTY OF THE HEAT EQUATION
13
Theorem 4. (A mean-value property for the heat equation). Let u ∈
C12 (UT ) solve the heat equation. Then
"
1
|y|2
u(x, t) = n
u(y, s) 2 dyds,
4r
s
E(x,t;r)
(4)
for each E(x, t; r) ⊂ UT .
Proof.
1. An useful identity:
"
|y|2
dyds = 4,
2
E(1) s
where E(1) = E(0, 0; 1).
Indeed,
"
E(1)
|y|2
dyds =
s2
=
=
=
=
=
=
=
=
Z
0
Z
1
ds
|y|2 dy
1 s2
1
2
− 4π
|y| ≤−2ns ln −4πs
1/2
1
Z
Z 0
ds [−2ns ln −4πs ]
nα(n)rn−1+2 dr
1
s
0
− 4π
"
# n+2
Z 0
2
nα(n)
1
1
(−s)
ds
2π
ln
n + 2 − 4π1 (−s)2
4π (−s)
! n+2
n+2 Z 1
2
4π
nα(n)(2n) 2
1
n−2
s 2 ln
ds
n+2
4πs
0
! n−2
!
n+2 Z 0
1 −s 2
nα(n)(2n) 2
1
n+2
−s
e
· s 2 · − e ds
n+2
4π
∞ 4π
Z ∞
n+2
1
nα(n)(2n) 2
n+4
n
s 2 −1 e− 2 s ds
·
n/2
n+2
(4π)
0
Z ∞ ! n+4
n+2
nα(n)(2n) 2
1
2 2 n+4
t 2 −1 e−t dt
·
n+2
(4π)n/2 0 n
n
8
·Γ +2
(n + 2)πn/2
2
n
n
8
πn/2
·
·
+
1
Γ
+
1
(n + 2)πn/2 Γ n + 1
2
2
2
= 4.
(5)
14
ZUJIN ZHANG
2. We now prove (4). Without loss of generality, we may assume
that (x, t) = (0, 0). Write E(r) = E(0, 0; r) and set
"
1
|y|2
u(y, s) 2 dyds
ϕ(r) ≡ n
r
s
E(r)
"
2
2 |y|
=
u(ry, r s) 2 dyds.
s
E(1)
Then
"
#
|y|2
|y|2
ϕ (r) =
dyds
y · Dy u 2 + 2rD s u
s
s
E(1)
#
" "
1
|y|2
|y|2
y · Dy u 2 + 2D s u
= n+1
dyds
r
s
s
E(r)
"
0
≡ A + B.
Next, we calculate B as
"
1
|y|2
B = n+1
2D s u dyds
r
s
E(r)
"
4
= n+1
D s uDy ϕ · ydyds (2)
r
E(r)
Z
"
4
4n
= − n+1
y · D s Dy uϕdyds − n+1
D s uϕdyds
r
r
E(r)
E(r)
integration by part w.r.t. y
"
#)
" (
4
n
|y|2
y · Dy u − − 2 dyds
= n+1
r
2s 4s
E(r)
"
4n
D s uϕdyds integration by part w.r.t. s and (3)
− n+1
r
E(r)
" 4
n
= −A + n+1
− y · Dy u − n∆y uϕ dyds D s u − ∆y u = 0
r
2s
E(r)
= −A integration by part w.r.t. y and (2) .
Hence,
"
ϕ(r) = lim ϕ(t) = lim
t→0+
t→0+
u(ry, r2 s)
E(1)
|y|2
dyds
s2
MEAN-VALUE PROPERTY OF THE HEAT EQUATION
"
=
u(0, 0)
E(1)
15
|y|2
dyds = 4u(0, 0).
s2
The proof of the mean-value property of the heat equation is thus
completed.
Acknowledgements. The author would like to thank Dr. Zhang at
Sun Yat-sen University, who are still on his road to Alexandrov geometry, and who showed me
\usepackage{palatino}
to display with the font here.
REFERENCES
[1] L.C. Evans, Weak convergence methods for nonlinear partial differential
equations. CBMS Regional Conference Series in Mathematics, 74. American
Mathematical Society, 1990.
D EPARTMENT OF M ATHEMATICS , S UN YAT- SEN U NIVERSITY , G UANGZHOU ,
510275, P.R. C HINA
E-mail address: [email protected]