31
2.4 Harmonic functions
2.4
Harmonic functions
Let ⌦ ⇢ Rn be an open connected region. A real valued function ' 2 C 2 (⌦) is said to
be harmonic if
' = 0.
The theory of harmonic functions is the same as the theory of conservative vector fields
with zero divergence. Indeed, for any vector field F in a connected region ⌦ ⇢ Rn one
has that one has curlF = 0 and divF = 0 if and only if there exists a harmonic potential
' making F = r'. An example that we already mentioned in the introduction is that
of an insulated region. Imagine a thin uniform metal plate that is insulated so no heat
can enter or escape. Some time after a given temperature distribution is maintained
along the edge of the plate, the temperature distribution inside the plate will reach a
steady-state, that will be given by a harmonic function '.
Throughout this section we write Br (x) ⇢ Rn for the ball of radios r centred at x 2 ⌦
and Sr (x) ⇢ Rn for the corresponding sphere. We set
cn := vol(S1 (0))
Theorem 7 (Mean value Theorem). Let ⌦ ⇢ Rn and let ' 2 C 2 (⌦) be a harmonic
function. Then, for each x 2 ⌦ and r > 0 such that B r (x) ⇢ ⌦ one has
Z
1
'(x) = n 1
'(y) d (y).
(2.4)
r
cn Sr (x)
Proof. Without loss of generality assume that x = 0. By Green’s identities, for any
U ⇢ Rn open and , 2 C 1 (U ),
Z
Z
(
) dx =
( @⌫
@⌫ ) d .
(2.5)
U
@U
For 0 < " < r set U := Br (0)\B" (0). Define := ' and (y) := |y|2 n whenever
n 6= 2 and (y) = log(|y|) when n = 2. Let us treat the case n 6= 2 (the other is done
similarly). It turns out that
= 0 on ⌦
and
@
|
= (2
@⌫ Sr (0)
n)r1
n
and
@
|
=
@⌫ S" (0)
(2
n)"1
n
.
Then,
0=
=
Z
('
Z⌦
') dx
( @⌫ ' '@⌫ ) d
Z
n
@⌫ ' d
"2
Sr (0)
= r2
Sr (0)
n
Z
(2
Z
( @⌫ '
'@⌫ ) d
S" (0)
@⌫ ' d
Z
1 n
n)r
S" (0)
' d + (2
Sr (0)
n)"
1 n
Z
'd
S" (0)
32
Laplacian in euclidean spaces
It is also clear from (2.5) that if we pick U := Br (0) or U := B" (0),
then
Z
Z
@⌫ ' d = 0
and
@⌫ ' d = 0.
Sr (0)
:= 1 and
:= ',
S" (0)
Therefore,
0=
r1
n
Z
' d + "1
n
Sr (0)
Z
'd .
S" (0)
By continuity of ',
1
rn
1c
n
Z
'(y) d (y) =
Sr (0)
1
"n
1c
n
Z
'(y) d (y)
S" (0)
!
"!0
'(0).
A function ' that satisfies (2.4) is said to have the mean value property. Satisfying
the mean value property is equivalent to
'(x) =
n
n
r cn
Z
'(y) dy.
(2.6)
Br (x)
Indeed, equation (2.6) follows from integrating '(x)rn 1 with respect to r. Equation
(2.4) follows from di↵erentiating '(x)rn with respect to r.
In addition, the mean value property is also equivalent to satisfying
'(x) =
1
cn
Z
'(x + rw) dS(w).
(2.7)
S1 (0)
where dS is the area measure on the unit sphere. This follows easily from performing
the change of variables y = x + r!.
Next we prove a converse to the Mean value Theorem.
Theorem 8. Let ⌦ ⇢ Rn and ' 2 C(⌦) satisfy the mean value property. Then ' is
smooth and harmonic in ⌦.
Corollary 9. Harmonic functions are smooth.
Proof of Theorem 8. Let Ru be a Friederich’s mollifier. That is, u 2 C01 (B1 (0)) is a
radial function satisfying B1 (0) u(x)dx = 1. For " > 0 define
u" (y) :=
1 ⇣y ⌘
u
.
"n
"
We will prove that '(x) = u" ⇤ '(x) for x 2 ⌦ with with 0 < " < dist(x, @⌦). Since u" is
smooth, it will follow that ' is smooth. In what follows, since u is radial, we introduce
33
2.4 Harmonic functions
v(r) := u(rw) for r 2 [0, 1) and w 2 S1 (0).
Z
u" ⇤ '(x) =
'(y)u" (y x)dy
Z⌦
=
'(x + y)u" (y)dy
⌦Z
⇣y ⌘
1
= n
'(x + y)u
dy
" B" (0)
"
Z
=
'(x + "y)u(y)dy
=
=
Z
Z
B1 (0)
1Z
0
'(x + "rw)u(rw)rn
S1 (0)
1
v(r)r
0
= '(x)cn
= u(x).
Z
n 1
1
Z
1
dS(w) dr
'(x + "rw) dS(w) dr
S1 (0)
v(r)rn
1
dr
0
It remains to show that ' is harmonic. Since ' is continuous, we deduce that
from the fact that for all r > 0
Z
Z
'(y) dy = =
@⌫ '(y)dS(y)
Br (x)
Sr (x)
Z
@
= rn 1
'(x + rw)dS(w)
@r S1 (0)
@
= rn 1 (cn '(x))
@r
= 0.
'=0
Harmonic functions are analytic (Exercise).
1. Fix x 2 Rn and let R > 0. Show that if ' 2 C 2 (BR (x)) \ C(BR (x)) is harmonic,
then
n
|@xi '(x)|  k'kL1 (BR (x)) .
R
Hint: Prove and use that @xi ' is harmonic.
2. Let ', x and R as in the previous part. For m 2 N, prove by induction that there
exists a constant C > 0 independent of m, n and R such that
|@ ↵ '(x)| 
nm C m 1 m!
Rm
for any multi-index ↵ with |↵| = m.
3. Prove by Taylor expansion that any harmonic function ' on ⌦ ⇢ Rn is analytic.
34
Laplacian in euclidean spaces
Theorem 10 (The Maximum Principle). Assume ⌦ ⇢ Rn is connected and open. If '
is harmonic and real-valued on ⌦, then
either
'(x) < sup '
⌦
8x 2 ⌦,
or
' = sup '.
⌦
Proof. Consider the set
A := {x 2 ⌦ : '(x) = sup '}.
⌦
The set A is clearly closed in A. The set A is also open. Indeed, if '(x) = sup⌦ ' then
'(y) = sup⌦ ' for all y in a ball centred at x for otherwise the Mean Value Theorem
would lead to a contradiction. Since A is both open and closed in ⌦ we conclude that
A = ⌦ or A = ;.
Theorem 11. Suppose ⌦ ⇢ Rn is open, and ⌦ is compact. If ' is harmonic and
real-valued on ⌦, and continuous on ⌦, then the maximum value of ' is achieved on
@⌦.
Proof. If the maximum is achieved at an interior point, the ' must be constant on the
connected component of ⌦ that contains such point, and therefore the maximum is also
achieved at the border.
Theorem 12. Suppose ⌦ is compact and that ',
up to @⌦. If '|@⌦ = |@⌦ , then
'=
are harmonic on ⌦ and continuous
on ⌦.
Proof. Consider the functions 1 = '
and 2 =
'. Both of them are harmonic
and equal to zero when restricted to the boundary. The result follows from the fact that
their maximums are achieved a the boundary.