Math 212b (Spring 2014) Yum-Tong Siu
1
WEAK DERIVATIVES
Riemannian manifold M with the Riemannian metric
∑nFor a compact
j
k
g
(x)dx
⊗
dx
, we introduce the Hilbert space L2 (M, p) of all square
j,k=1 jk
integrable p-forms on M and the Hilbert space L21 (M, p) of all p-forms on M
with square integrable derivatives of order ≤ 1. The Hilbert space L2 (M, p) is
defined abstractly as the completion of the vector space of all smooth p-forms
φ=
∑
1
p! 1≤j ,··· ,j
1
φj1 ,··· ,jp dxj1 ∧ · · · ∧ dxjp
p ≤n
on M with the inner product (φ, φ)
L2 (M,p)
1
p!
∑
1≤j1 ,··· ,jp ≤n,
1≤k1 ,··· ,kp ≤n
, which is defined as
∫
g j1 k1 · · · g jp kp φj1 ,··· ,jp φk1 ,··· ,kp
M
√
det (gjk )1≤j,k≤n dx1 ∧ · · · ∧ dxn .
The Hilbert space L21 (M, p) is defined abstractly as the completion of the
vector space of all smooth p-forms φ on M the inner product (φ, φ) 2
,
L1 (M,p)
which is defined as the sum of φ with itself as the sum of (φ, φ)
L2 (M,p)
1
p!
∫
∑
g µν g j1 k1 · · · g jp kp ∇µ φj1 ,··· ,jp ∇ν φk1 ,··· ,kp
1≤µ,ν≤n, 1≤j1 ,··· ,jp ≤n,
1≤k1 ,··· ,kp ≤n
M
and
√
det (gjk )1≤j,k≤n dx1 ∧· · ·∧dxn .
Elements of the abstractly defined Hilbert space L2 (M, p) can be more
concretely characterized as the vector space of all global p-forms
φ=
∑
1
p! 1≤j ,··· ,j
1
φj1 ,··· ,jp dxj1 ∧ · · · ∧ dxjp
p ≤n
on M such that each local coefficient φj1 ,··· ,jp of φ is locally square integrable
on M . However, in order to more concretely characterize the elements of
the abstractly defined Hilbert space L21 (M, p), we need the notion of weak
derivatives for the local coefficients φj1 ,··· ,jp of φ.
We discuss here the notion of weak derivatives of a local L2 function. The
key result is the 1944 theorem of Friedrichs in
Math 212b (Spring 2014) Yum-Tong Siu
2
Kurt Otto Friedrichs, The identity of weak and strong extensions of differential operators.
Trans. Amer. Math. Soc. 55 (1944), 132 – 151,
which states that for a local L2 function f to have a local L2 function as a
weak derivative of a local L2 function (or more generally as the result Lf
of applying a first-order differential operator L with smooth coefficients) is
equivalent to be the limit of a sequence of smooth functions fν with Lfν
converging in L2 locally to Lf .
This means that elements φ of the abstractly defined Hilbert space L21 (M, p)
are characterized by being global p-forms
φ=
∑
1
p! 1≤j ,··· ,j
1
φj1 ,··· ,jp dxj1 ∧ · · · ∧ dxjp
p ≤n
on M such that each local coefficient φj1 ,··· ,jp of φ is locally square integrable
on M with each of its covariant derivatives ∇ν φj1 ,··· ,jp taken in the weak
sense being locally integrable. The weak derivative τ (t) of a locally integrable
function σ(t) of a single real variable t means a locally integrable function
such that
∫
∫
τ (t)ψ(t)dt = − σ(t)ψ ′ (t)dt
for any smooth function ψ(t) with compact support in the domain where
the above two integrations are performed. We would like to remark that if
the locally integrable function σ(t) has compact support on R and if τ (t) is
its weak derivative, then for any smooth function ψ(t) on R with compact
support,
d
(ψ ∗ σ) = ψ ∗ τ
dt
holds, because
∫
∫
′
′
(ψ ∗ σ) (t) =
ψ (s)σ(t − s)ds =
ψ(s)τ (t − s)ds,
s∈R
s∈R
where the last identity comes from the assumption that τ is the weak derivative of σ.
∂
Theorem of Friedrichs. Suppose L = a(x) ∂x
+ b(x) is a first-order differential
operator with smooth
a(x), b(x) with compact support on Rn .
∑ncoefficients
∂
∂
Here a(x) ∂x means j=1 aj (x) ∂xj . Suppose u(x) is an L2 function on Rn
Math 212b (Spring 2014) Yum-Tong Siu
3
with compact support such that Lu is L2 when the partial differentiation
with respect to the coordinates x1 , · · · , xn are taken in the weak sense. Let
χ(x)
be a nonnegative function supported on the unit open ball of Rn with
∫
χ(x) = 1. Let χϵ (x) = ϵ1n χ( ϵxn ). Then χϵ ∗ u → u in L2 norm as ϵ → 0
Rn
and χϵ ∗ Lu → Lu in L2 norm as ϵ → 0.
Proof. The statement that χϵ ∗ Lu → Lu in L2 norm as ϵ → 0 is a simple of
the continuous version of the triangle inequality, because
∫
χε (y)u(x − y)
(χϵ ∗ u)(x) =
y∈Rn
∫
and
(χϵ ∗ u)(x) − u(x) =
χε (y) (u(x − y) − u(x))
y∈Rn
so that from the continuous version of the triangle inequality
∫
∥(χϵ ∗ u) − u∥L2 (Rn ) =
χε (y) ∥Transly u − u∥L2 (Rn )
y∈Rn
(where Transly u means the translate of u by y with (Transly u)(x) = u(x−y))
we can use
lim ∥Transly u − u∥L2 (Rn ) = 0
y→0
to conclude that
lim ∥(χϵ ∗ u) − u∥L2 (Rn ) = 0.
ε→0
To show that
lim ∥(χϵ ∗ Lu) − Lu∥L2 (Rn ) = 0,
ε→0
it suffices to show that χϵ ∗ Lu − L(χϵ ∗ u) approaches 0 in L2 norm as ϵ → 0.
This is clearly true when u belongs to the dense subset of smooth functions.
Introduce the operator Tε defined by Tε u = χϵ ∗ Lu − L(χϵ ∗ u). The
key point of this proof is that by the standard “3 ε−argument” it suffices to
show that the collection of operators {Tε }0<ε≤ε0 is bounded in L2 norm (by
some positive number Aε0 depending only on ε0 but independent of ε) when
u belongs to a set bounded in L2 norm. The reason is that for any given
η > 0, if v is in the dense set (of smooth functions) with ∥u − v∥L2 (Rn ) < η,
then
∥Tε u∥L2 (Rn ) ≤ ∥Tε (u − v)∥L2 (Rn ) + ∥Tε v∥L2 (Rn ) ≤ Aε0 η + ∥Tε v∥L2 (Rn )
Math 212b (Spring 2014) Yum-Tong Siu
4
and we can choose ε sufficiently small to get
∥Tε v∥L2 (Rn ) < η.
Note that the bound of the inequality in the argument is only the sum of two
terms instead of the sum of three term, because the normal “3 ε−argument”
involves Tε u → T0 u, but here T0 is the zero operator so that the term T0 (u−v)
is not present. In words, this key argument uses the statement that a sequence
of operators of a Hilbert space strongly converges everywhere if it is uniformly
bounded in norm and strongly converges at every point of a dense subset.
Strong convergence of Sν → S means Sν f → Sf for every element f of the
Hilbert space and is so-called to distinguish it from weak convergence which
means (Sν f, g) → (Sf, g) for any pair of elements f, g of the Hilbert space
and to distinguish it from convergence in norm which means ∥Sν − S∥ → 0.
Now we verify that the operators Tε defined by Tε u = χϵ ∗ Lu − L(χϵ ∗ u)
are uniformly bounded in norm for 0 < ε ≤ ε0 . The zero-order part b(x)
of L clearly has bounded contribution. So we can assume without loss of
generality that b(x) = 0. Then
(
)
∂u
∂
χϵ ∗ Lu − L(χϵ ∗ u) = χϵ ∗ a
− a (χϵ ∗ u)
∂x
∂x
(
)
(
)
∂
∂
∂a
= χϵ ∗
(au) − χϵ ∗ u
−a
χϵ ∗ u
∂x
∂x
∂x
(
)
(
)
)
(
∂
∂
∂a
=
χϵ ∗ (au) − χϵ ∗ u
−a
χϵ ∗ u .
∂x
∂x
∂x
The purpose of the above manipulation is to get rid of all differentiation of
u by using the property of weak derivatives. Clearly the second term on the
right-hand side is bounded. So we can drop it. We have
((
)
(
))
∂
∂
χϵ ∗ (au) − a
χϵ ∗ u
(x)
∂x
∂x
)
(
)
)
∫ ((
∂
∂
χϵ (y)a(x − y)u(x − y) − a(x)
χϵ (y) ∗ u(x − y) dy
=
∂y
∂y
(
)
∫
∂
=
χϵ (y) (a(x − y) − a(x)) u(x − y)dy.
∂y
|y|<ϵ
Math 212b (Spring 2014) Yum-Tong Siu
5
In the last integral |a(x − y) − a(x)| ≤ Cϵ because |y| < ϵ. We have
∂ χϵ ≤ C ′ 1 .
∂y ϵn+1
Hence
((
∂
χϵ
∂x
)
(
))
∫
∂
′′ 1
|u(x − y)|dy
χϵ ∗ u
(x) ≤ C n
∂x
ϵ |y|<ϵ
∗ (au) − a
and the L2 norm of
(
∂
χϵ
∂x
is bounded by
1
C n
ϵ
′′′
)
(
∗ (au) − a
∫
∂
χϵ ∗ u
∂x
)
)
(∫
|u(x − y)| dx dy
2
|y|<ϵ
which is bounded if the L2 norm of u is bounded. Q.E.D.
© Copyright 2024 ExpyDoc
