The Mollifier Theorem
Definition of the Mollifier
The function
Tx =
K exp
−1  if |x| < 1
1 − |x| 2
,
x ∈ Rn
if |x| ≥ 1
0
where the constant K is chosen such that ∫ n Txdx = 1, is a test function on R n . Note
R
that Tx vanishes, together with all its derivatives as |x| → 1 − , so Tx is infinitely
differentiable and has compact support. The graph of Tx is sketched in the following
figure.
The Mollifier Function
.For n = 1 and  > 0, let
S  x = 1 T x
P ε x = T x .
and
Then
S  x ≥ 0
and
P ε x ≥ 0
for all x
S  x = 0
and
P ε x = 0
for | x| > 
∫R S  x dx = 1
∀ > 0,
∫R P  x dx → 0
as  → 0,
S  0 → +∞ as  → 0,
P  0 = K/e
∀ > 0,
Evidently, S  x becomes thinner and higher as  tends to zero but the area under the
graph is constantly equal to one. On the other hand, P ε x has constant height but grows
thinner as  tends to zero. These test functions can be used as the ”seeds” from which an
infinite variety of other test functions can be constructed by using a technique called
regularization which we will now describe.
For n ≥ 1 we have
S  x = 1n T x

and
P ε x = T x .
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For U a bounded open set in R n , and for u ∈ L 1loc U, define for any  > 0 and any x ∈ U 
= x ∈ U : distx, ∂U > ,
J  ux = ∫
|x−y|≤ 
=∫
|z|≤ 
=∫
|z|≤ 1
S  x − y uy dy
1.1a
S  z ux − z dz
1.1b
S 1 z ux − z dz.
1.1c
We refer to J  ux as the mollified ux. This mollified function, J  ux, is a smoothed
version of the original function, ux.
Properties of the Mollifier
Note first that J  ux is infinitely differentiable; i.e., for any  > 0 and any x ∈ U  , it is clear
from (1.1a) that
J  ux + e⃗i  − J  ux
S  x + e⃗i − y − S  x − y
=∫
uy dy



|x−y|≤
i.e.,
J  ux + e⃗i  − J  ux
→∫
∂ x i S  x − y uy dy

|x−y|≤ 
as  → 0.
Since S  x is infinitely differentiable, it follows that J  ux is infinitely differentiable on the
open set U  .
It is evident from (1.1a) that for 1 ≤ p < ∞,  > 0, and x ∈ U  ,
J  ux = ∫
|x−y|≤ 
S  x − y 1−1/p S  x − y 1/p uy dy.
Then, using Holder’s inequality, we get
| J  ux| p =
∫|x−y|≤  S  x − y dy
and since ∫ S  x dx = 1
R
p−1
∫|x−y|≤  S  x − y| uy| p dy
∀ > 0,
∫V |J  ux| p dx ≤ ∫V ∫|x−y|≤  S  x − y| uy| p dy dx
=∫
W
| uy| p ∫
|x−y|≤ 
S  x − ydx dy = ∫
W
| uy| p dy
for open sets W = U  , and V = W  . This result is just that assertion that
|| J  u|| L p V ≤ || u|| L p W for V ⊂⊂ W ⊂⊂ U
1.2
Next, use (1.1c) to write
J  ux − ux = ∫
|z|≤ 1
S 1 z  ux − z − uz dz.
If the function u = ux is, in fact, continuous on U, then this last result shows that
max |J  ux − ux| ≤ max |  ux − z − uz | → 0 as  → 0;
V̄
V̄
1.3
i.e., J  ux converges uniformly to ux for x ∈ V̄ when ux is continuous on .U.
For u ∈ L ploc U, W = U  , and arbitrary δ > 0, use the fact that the continuous functions
2
are dense in L p W to choose v ∈ CW such that
|| u − v|| L p W ≤ δ.
Then for V = W  ,
|| J  u − u|| L p V ≤ || J  u − J  v|| L p V + || J  v − v|| L p V + || v − u|| L p V
≤ || u − v|| L p W + || J  v − v|| L p V + || v − u|| L p W ≤ 2δ + || J  v − v|| L p V
It follows now from (1.3) that for u ∈ L ploc U ,
∀V ⊂⊂ U,
|| J  u − u|| L p V → 0 as  → 0
1.4
We can summarize these results in the following,
Theorem (Local Approximation) Suppose U is open and bounded in R n , 1 ≤ p < ∞, and for
̇ > 0, let U  denote the subset x ∈ U : distx, ∂U > .
u ∈ L ploc U implies J ε u ∈ C ∞ U ε 
(a) For every ε > 0,
(b) (i) u ∈ CU implies u  converges to u uniformly on compact subsets of U; i.e.,
‖J  u − u‖ CV̄  = max |J ε ux − ux|  0 for all V ⊂⊂ U
̄
V
(ii) u  converges to u in
Jεu
L p V
≤
L ploc U;
u
i.e., u ∈ L ploc U implies that for all V ⊂⊂ W ⊂⊂ U,
L p W
and
Jεu − u
L p V
 0 as ε  0
(c) u  converges to u in W k,p
loc U;
Result (c) follows from (b) by induction.
Corollary (Global Approximation) Suppose U has a smooth boundary, and 1 ≤ p < ∞.
u ∈ L p U implies J ε u ∈ C ∞ U ∩ L p U.
(a) For every ε > 0,
(b) u ∈ L p U implies that
Jεu
L p U
≤
u
L p U
and
Jεu − u
L p U
 0 as ε  0
(c) u ∈ W k,p U implies that there exists functions φ m  ∈ C ∞ U ∩ W k,p U such that
||φ m − u|| k,p → 0
as m → ∞.
The proof of the corollary makes use of a partition of unity (see theorem 2 pg 251 in
Evans).
Weak Equals Strong
For U a bounded open set in R n , we define v = vx to be the weak derivative of order α, of
u = ux, x ∈ U if
∫U ux ∂ α φx dx = −1 |α| ∫U vx φx dx for all φ ∈ C ∞c U
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Similarly, we define v = vx to be the strong L p −derivative of order α, of
u = ux, x ∈ U if
for any V ⊂⊂ U, there exists a sequence φ n  ∈ C ∞c U such that
∫V |φ n − u| p dx → 0 and ∫V | ∂ α φ n − v| p dx → 0, as n → ∞.
Using mollifiers, we can show that these two notions are equivalent.
Suppose first that v = vx is the weak derivative of order α, of u = ux. Then, since
S  ∈ C ∞c U,
∂ α J  ux = ∫
∂ αx S  x − y uy dy = −1 |α| ∫
∂ αy S  x − y uy dy
|x−y|≤ 
|x−y|≤ 
=∫
|x−y|≤ 
S  x − y vy dy = J  vx
(by definition of weak derivative)
Now apply 1.4 to write
∫V |J  u − u| p dx → 0 and ∫V | ∂ α J  u − v| p dx = ∫V | J  v − v| p dx → 0, as n → ∞.
Thus every weak derivative is a strong L p −derivative.
Conversely, suppose v = vx is the strong L p −derivative of order α, of u = ux with
∫V |φ n − u| p dx → 0 and ∫V | ∂ α φ n − v| p dx → 0, as n → ∞,
for arbitrary V ⊂⊂ U, and φ n  ∈ C ∞c U. Then for any ψ ∈ C ∞c U,
∫V u − φ n  ∂ α ψ dx = ∫V u ∂ α ψ dx − ∫V φ n ∂ α ψ dx
= ∫ u ∂ α ψ dx − −1 |α| ∫ ∂ α φ n ψ dx
V
V
= ∫ u ∂ α ψ dx − −1 |α| ∫ v ψ dx + −1 |α| ∫ v − ∂ α φ n  ψ dx
V
V
V
Then it follows that
∫V u ∂ α ψ dx − −1 |α| ∫V v ψ dx ≤ C 1 ∫V |φ n − u| p dx + C 2 ∫V |∂ α φ n − v| p dx
which implies that every strong L p −derivative is a weak derivative.
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Weyl’s Lemma
Weyl’s lemma is a famous result that asserts that for U a bounded open set in R n , if
u = ux is harmonic in U, (i.e., u ∈ C 2 U and ∇ 2 ux = 0, x ∈ U ) then ux is infinitely
differentiable in U.
To see why this result is true, recall that every harmonic function has the mean value
property. That is,
ux = ∫
∀x ∈ U  , r < ,
Then
J  ux = ∫
|x−y|≤ 
uy dŜy =
1
∫
uy dSy.
nr n−1 A n ∂B r x
x−y
S  x − y uy dy = 1n ∫
T

 |x−y|≤ 

= 1n ∫ T r
 0
= ux ∫
∂B r x
B  0
uy dy

n
∫∂B x uy dSy dr = ux ∫ 0 nA
T r r n−1 dr
n
r
S  y dy = ux.
But this says that ∀ > 0, ∀x ∈ U  , J  ux = ux. Since J  ux is infinitely differentiable on
U, it follows that ux is infinitely differentiable on U although u need not even be
continuous on the closure, Ū.
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