III Analysis of Partial Differential Equations

3Function spaces

3.3 Approximation of functions in Sobolev spaces

It would be nice if we could approximate functions in

k,p

(

) with something

more tractable. For example, it would be nice if we could approximate them by

smooth functions, so that the weak derivatives are genuine derivatives. A useful

trick to improve regularity of a function is to convolve with a smooth mollifier.

Definition (Standard mollifier). Let

η(x) =

(

1/(|x|

−1)

|x| < 1

0 |x| ≥ 1

where C is chosen so that

η(x) dx = 1.

One checks that this is a smooth function on R

, peaked at x = 0.

For each ε > 0, we set

(x) =





Of course, the pre-factor of

is chosen so that

is appropriately normalized.

We call η

the standard mollifier, and it satisfies supp η

⊆ B

(0).

We think of these η

as approximations of the δ-function.

Now suppose U ⊆ R

is open, and let

= {x ∈ U : dist(x, ∂U) > ε}.

Definition

(Mollification)

f ∈ L

loc

(

), we define the mollification

→

R by the convolution

= η

∗ f.

In other words,

(x) =

(x − y)f(y) =

(x)

(x − y)f(y) dy.

Thus,

is the “local average” of

around each point, with the weighting

given by

. The hope is that

will have much better regularity properties

than f.

Theorem. Let f ∈ L

loc

(U). Then

(i) f

∈ C

∞

(ii) f

→ f almost everywhere as ε → 0.

(iii) If in fact f ∈ C(U), then f

→ f uniformly on compact subsets.

(iv)

If 1

≤ p < ∞

and

f ∈ L

loc

(

), then

→ f

loc

(

), i.e. we have

convergence in L

on any V b U.

In general, the difficulty of proving these approximation theorems lie in what

happens at the boundary

Lemma. Assume u ∈ W

k,p

(U) for some 1 ≤ p < ∞, and set

= η

∗ u on U

Then

(i) u

∈ C

∞

) for each ε > 0

(ii) If V b U, then u

→ u in W

k,p

(V ).

Proof.

(i) As above.

(ii) We claim that

= η

∗ D

for |α| ≤ k in U

To see this, we have

(x) = D

(x − y)u(y) dy

(−1)

|α|

(x − y)u(y) dy

For a fixed

x ∈ U

(

x − ·

)

∈ C

∞

(

), so by the definition of a weak

derivative, this is equal to

(x − y)D

u(y) dy

= η

∗ D

It is an exercise to verify that we can indeed move the derivative past the

integral.

Thus, if we fix

V b U

. Then by the previous parts, we see that D

→

u in L

(V ) as ε → 0 for |α| ≤ k. So

− uk

k.p

(V )

|α|≤k

− D

(V )

→ 0

as ε → 0.

Theorem

(Global approximation)

Let 1

≤ p < ∞

, and

U ⊆ R

be open and

bounded. Then C

∞

(U) ∩ W

k,p

(U) is dense in W

k,p

(U).

Our main obstacle to overcome is the fact that the mollifications are only

defined on U

, and not U.

Proof. For i ≥ 1, define



x ∈ U | dist(x, ∂U) >



= U

i+3

−

i+1

= U

i+4

−

We clearly have

∞

i=1

, and we can choose

b U

such that

∞

i=0

Let

{ζ

}

∞

i=0

be a partition of unity subordinate to

}

. Thus, we have

0 ≤ ζ

≤ 1, ζ

∈ C

∞

) and

∞

i=0

= 1 on U.

Fix δ > 0. Then for each i, we can choose ε

sufficiently small such that

= η

∗ ζ

satisfies supp u

⊆ W

and

− ζ

k.p

(U)

= ku

− ζ

k.p

)

≤

i+1

Now set

v =

∞

i=0

∈ C

∞

(U).

Note that we do not know (yet) that

v ∈ W

k.p

(

). But it certainly is when we

restrict to some V b U.

In any such subset, the sum is finite, and since u =

∞

i=0

u, we have

kv −uk

k,p

(V )

≤

∞

i=0

− ζ

k.p

(V )

≤ δ

∞

i=0

−(i+1)

= δ.

Since the bound

does not depend on

, by taking the supremum over all

we have

kv −uk

k.p

(U)

≤ δ.

So we are done.

It would be nice for

∞

(

) to be dense, instead of just

∞

(

). It turns out

this is possible, as long as we have a sensible boundary.

Definition

(

k,δ

boundary)

Let

U ⊆ R

be open and bounded. We say

∂U

k,δ

if for any point in the boundary

p ∈ ∂U

, there exists

r >

0 and a function

γ ∈ C

k,δ

(

n−1

) such that (possibly after relabelling and rotating axes) we have

U ∩ B

(p) = {(x

, x

) ∈ B

(p) : x

> γ(x

)}.

Thus, this says our boundary is locally the graph of a C

k,δ

function.

Theorem

(Smooth approximation up to boundary)

Let 1

≤ p < ∞

, and

U ⊆ R

be open and bounded. Suppose

∂U

0,1

. Then

∞

(

)

∩ W

k,p

(

) is

dense in W

k,p

(U).

Proof.

Previously, the reason we didn’t get something in

∞

(

) was that we

had to glue together infinitely many mollifications whose domain collectively

exhaust

, and there is no hope that the resulting function is in

∞

(

). In the

current scenario, we know that U locally looks like

The idea is that given a

defined on

, we can shift it downwards by some

It is a known result that translation is continuous, so this only changes

by a

tiny bit. We can then mollify with a

¯ε < ε

, which would then give a function

defined on U (at least locally near x

So fix some

∈ ∂U

. Since

∂U

0,1

, there exists

r >

0 such that

γ ∈ C

0,1

n−1

) such that

U ∩ B

) = {(x

, x

) ∈ B

) | x

> γ(x

)}.

Set

V = U ∩ B

r/2

Define the shifted function u

to be

(x) = u(x + εe

Now pick ¯ε sufficiently small such that

ε,¯ε

= η

¯ε

∗ u

is well-defined. Note that here we need to use the fact that

∂U

0,1

. Indeed,

we can see that if the slope of ∂U is very steep near a point x:

then we need to choose a

¯ε

much smaller than

. By requiring that

is 1-H¨older

continuous, we can ensure there is a single choice of

¯ε

that works throughout

As long as ¯ε is small enough, we know that v

ε,¯ε

∈ C

∞

(

V ).

Fix δ > 0. We can now estimate

ε,˜ε

− uk

k.p

(V )

= kv

ε,˜ε

− u

+ u

− uk

k,p

(V )

≤ kv

ε,˜ε

− u

k,p

(V )

+ ku

− uk

k.p

(V )

Since translation is continuous in the

norm for

p < ∞

, we can pick

ε >

such that

− uk

k.p

(V )

. Having fixed such an

, we can pick

˜ε

so small

that we also have kv

ε,˜ε

− u

k.p

(V )

The conclusion of this is that for any

∈ ∂U

, we can find a neighbourhood

V ⊆ U

such that for any

u ∈ W

k,p

(

) and

δ >

0, there exists

v ∈ C

∞

(

V ) such that ku − vk

k,p

(V )

≤ δ.

It remains to patch all of these together using a partition of unity. By the

compactness of

∂U

, we can cover

∂U

by finitely many of these

, say

, . . . , V

We further pick a V

such that V

b U and

U =

[

i=0

We can pick approximations

∈ C

∞

(

) for

= 0

, . . . , N

(the

= 0 case is given

by the previous global approximation theorem), satisfying

− uk

k,p

)

≤ δ

Pick a partition of unity {ζ

}

i=0

U subordinate to {V

}. Define

v =

i=0

Clearly v ∈ C

∞

(

U), and we can bound

v −D

(U)



i=0

− D

i=0



(U)

≤ C

i=0

− uk

k.p

)

≤ C

(1 + N)δ,

where

is a constant that solely depends on the derivatives of the partition of

unity, which are fixed. So we are done.