Lecture Notes on Sobolev Spaces
Alberto Bressan
February 27, 2012
1 Distributions and weak derivatives
We denote by L1
loc(IR) the space of locally integrable functions f : IR → IR. These are the
Lebesgue measurable functions which are integrable over every bounded interval.
The support of a function φ, denoted by Supp(φ), is the closure of the set {x ; φ(x) = 0}
where φ does not vanish. By C∞
c (IR) we denote the space of continuous functions with compact
support, having continuous derivatives of every order.
Every locally integrable function f ∈ L1
loc(IR) determines a linear functional Λf : C∞
c (IR) → IR,
namely
Λf (φ)
.
=
IR
f(x)φ(x) dx . (1.1)
Notice that this integral is well defined for all φ ∈ C∞
c (IR) , because φ vanishes outside a
compact set. Moreover, if Supp(φ) ⊆ [a, b], we have the estimate
|Λf (φ)| ≤
b
a
|f(x)| dx φ C0 . (1.2)
Next, assume that f is continuously differentiable. Then its derivative
f (x) = lim
h→0
f(x + h) − f(x)
h
is continuous, hence locally integrable. In turn, f also determines a linear functional on
C∞
c (IR), namely
Λf (φ)
.
=
IR
f (x) φ(x) dx = −
IR
f(x) φ (x) dx . (1.3)
At this stage, a key observation is that the first integral in (1.3) is defined only if f (x) exists
for a.e. x, and is locally integrable. However, the second integral is well defined for every
locally integrable function f, even if f does not have a pointwise derivative at any point.
Moreover, if Supp(φ) ⊆ [a, b], we have the estimate
|Λf (φ)| ≤
b
a
|f(x)| dx φ C1 .
This construction can be performed also for higher order derivatives.
1
Definition 1.1 Given an integer k ≥ 1, the distributional derivative of order k of f ∈ L1
loc
is the linear functional
ΛDkf (φ)
.
= (−1)k
IR
f(x)Dk
φ(x) dx .
If there exists a locally integrable function g such that ΛDkf = Λg, namely
IR
g(x)φ(x) dx = (−1)k
IR
f(x)Dk
φ(x) dx for all φ ∈ C∞
c (IR),
then we say that g is the weak derivative of order k of f.
Remark 1.1 Classical derivatives are defined pointwise, as limits of difference quotients. On
the other hand, weak derivatives are defined only in an integral sense, up to a set of measure
zero. By arbitrarily changing the function f on a set of measure zero we do not affect its weak
derivatives in any way.
Example 1. Consider the function
f(x)
.
=
0 if x ≤ 0,
x if x > 0 .
Its distributional derivative is the map
Λ(φ) = −
∞
0
x · φ (x) dx =
∞
0
φ(x) dx =
IR
H(x) φ(x) dx ,
where
H(x) =
0 if x ≤ 0,
1 if x > 0 .
(1.4)
In this case, the Heaviside function H in (1.4) is the weak derivative of f.
Example 2. The function H in (1.4) is locally integrable. Its distributional derivative is the
linear functional
Λ(φ)
.
= −
IR
H(x) φ (x) dx −
∞
0
φ(x) dx = φ(0) .
This corresponds to the Dirac measure, concentrating a unit mass at the origin. We claim
that the function H does not have any weak derivative. Indeed, assume that, for some locally
integrable function g, one has
g(x)φ(x) dx = φ(0) for all φ ∈ C∞
c .
By the Lebesgue Dominated Convergence Theorem,
lim
h→0
h
−h
|g(x)| dx = 0.
2
Hence we can choose δ > 0 so that δ
−δ |g(x)| dx ≤ 1/2. Let φ : IR → [0, 1] be a smooth
function, with φ(0) = 1 and with support contained in the interval [−δ, δ]. We now reach a
contradiction by writing
1 = φ(0) = Λ(φ) =
IR
g(x)φ(x) dx =
δ
−δ
g(x)φ(x) dx ≤ max
x
|φ(x)|·
δ
−δ
|g(x)| dx ≤
1
2
.
Example 3. Consider the function
f(x)
.
=
0 if x is rational,
2 + sin x if x is irrational .
Clearly f is discontinuous at every point x. Hence it is not differentiable at any point. On the
other hand, the function g(x) = cos x provides a weak derivative for f. Indeed, the behavior
of f on the set of rational points (having measure zero) is irrelevant. We thus have
− f(x)φ (x) dx = − (2 + sin x) φ (x) dx = (cos x) φ(x) dx.
f
φ
0
1
1 b x1/3 2/3
1/2
Figure 1: The Cantor function f and a test function φ showing that g(x) ≡ 0 cannot be the weak
derivative of f.
Example 4. Consider the Cantor function f : IR → [0, 1], defined by
f(x) =



0 if x ≤ 0 ,
1 if x ≥ 1 ,
1/2 if x ∈ [1/3, 2/3] ,
1/4 if x ∈ [1/9, 2/9] ,
3/4 if x ∈ [7/9, 8/9] ,
· · ·
(1.5)
This provides a classical example of a continuous function which is not absolutely continuous.
We claim that f does not have a weak derivative. Indeed, let g ∈ L1
loc be a weak derivative of
f. Since f is constant on each of the open sets
] − ∞, 0[ , ]1, +∞[ ,
1
3
,
2
3
,
1
9
,
2
9
,
7
9
,
8
9
, . . .
3
we must have g(x) = f (x) = 0 on the union of these open intervals. Hence g(x) = 0 for
a.e. x ∈ IR. To obtain a contradiction, it remains to show that the function g ≡ 0 is NOT the
weak derivative of f. As shown in Fig. 1, let φ ∈ C∞
c be a test function such that φ(x) = 1
for x ∈ [0, 1] while φ(x) = 0 for x ≥ b. Then
g(x)φ(x) dx = 0 = 1 = − f(x)φ (x) dx .
1.1 Distributions
The construction described in the previous section can be extended to any open domain in
a multi-dimensional space. Let Ω ⊆ IRn be an open set. By L1
loc(Ω) we denote the space of
locally integrable functions on Ω. These are the measurable functions f : Ω → IR which are
integrable restricted to every compact subset K ⊂ Ω.
Example 5. The functions ex, and ln |x| are in L1
loc(IR), while x−1 /∈ L1
loc(IR). On the other
hand, the function f(x) = xγ is in L1
loc(]0, ∞[) for every (positive or negative) exponent γ ∈ IR.
In several space dimensions, the function f(x) = |x|−γ is in L1
loc(IRn) provided that γ < n.
One should keep in mind that the pointwise values of a function f ∈ L1
loc on a set of measure
zero are irrelevant.
By C∞
c (Ω) we denote the space of continuous functions φ : Ω → IR, having continuous partial
derivatives of all orders, and whose support is a compact subset of Ω. Functions φ ∈ C∞
c (Ω)
are usually called “test functions”. We recall that the support of a function φ is the closure
of the set where φ does not vanish:
Supp (φ)
.
= {x ∈ Ω ; φ(x) = 0}.
We shall need an efficient way to denote higher order derivatives of a function f. A multiindex
α = (α1, α2, . . . , αn) is an n-tuple of non-negative integer numbers. Its length is defined
as
|α|
.
= α1 + α2 + · · · + αn .
Each multi-index α determines a partial differential operator of order |α|, namely
Dα
f =
∂
∂x1
α1
∂
∂x2
α2
· · ·
∂
∂xn
αn
f .
Definition 1.2 By a distribution on the open set Ω ⊆ IRn we mean a linear functional
Λ : C∞
c (Ω) → IR such that the following boundedness property holds.
• For every compact K ⊂ Ω there exist an integer N ≥ 0 and a constant C such that
|Λ(φ)| ≤ C φ CN for every φ ∈ C∞ with support contained inside K. (1.6)
In other words, for all test functions φ which vanish outside a given compact set K, the value
Λ(φ) should be bounded in terms of the maximum value of derivatives of φ, up to a certain
order N.
4
Notice that here both N and C depend on the compact subset K. If there exists an integer
N ≥ 0 independent of K such that (1.6) holds (with C = CK possibly still depending on K),
we say that the distribution has finite order. The smallest such integer N is called the order
of the distribution.
Example 6. Let Ω be an open subset of IRn and consider any function f ∈ L1
loc(Ω). Then
the linear map Λf : C∞
c (Ω) → IR defined by
Λf (φ)
.
=
Ω
f φ dx , (1.7)
is a distribution. Indeed, it is clear that Λf is well defined and linear. Given a compact subset
K ⊂ Ω, for every test function φ with Supp(φ) ⊆ K we have the estimate
|Λf (φ)| =
K
f φ dx ≤
K
|f(x)| dx · max
x∈K
|φ(x)| ≤ C φ C0 .
Hence the estimate (1.6) holds with C = K |f| dx and N = 0. This provides an example of a
distribution of order zero.
The family of all distributions on Ω is clearly a vector space. A remarkable fact is that,
while a function f may not admit a derivative (in the classical sense), for a distribution Λ an
appropriate notion of derivative can always be defined.
Definition 1.3 Given a distribution Λ and a multi-index α, we define the distribution DαΛ
by setting
Dα
Λ(φ)
.
= (−1)|α|
Λ(Dα
φ) . (1.8)
It is easy to check that DαΛ is itself a distribution. Indeed, the linearity of the map φ →
DαΛ(φ) is clear. Next, let K be a compact subset of Ω and let φ is a test function with
support contained in K. By assumption, there exists a constant C and an integer N ≥ 0 such
that (1.6) holds. In turn, this implies
|Dα
Λ(φ)| = |Λ(Dα
φ)| ≤ C Dα
φ CN ≤ C φ CN+|α| .
Hence DαΛ also satisfies (1.6), with N replaced by N + |α|.
Notice that, if Λf is the distribution at (1.7) corresponding to a function f which is |α|-times
continuously differentiable, then we can integrate by parts and obtain
Dα
Λf (φ) = (−1)|α|
Λf (Dα
φ) = (−1)|α|
f(x)Dα
φ(x) dx = Dα
f(x)φ(x) dx = ΛDαf (φ) .
This justifies the formula (1.8).
1.2 Weak derivatives
For every locally integrable function f and every multi-index α = (α1, . . . , αn), the distribution
Λf always admits a distributional derivative DαΛf , defined acording to (1.8). In some cases,
one can find a locally integrable function g such that the distribution DαΛf coincides with
the distribution Λg. This leads to the concept of weak derivative.
5
Definition 1.4 Let f ∈ L1
loc(Ω) be a locally integrable function on the open set Ω ⊆ IRn and
let Λf be the corresponding distribution, as in (1.7). Given a multi-index α = (α1, . . . , αn), if
there exists a locally integrable function g ∈ L1
loc(Ω) such that DαΛf = Λg, i.e.
f Dα
φ dx = (−1)|α|
g φ dx for all test functions φ ∈ C∞
c (Ω), (1.9)
then we say that g is the weak α-th derivative of f, and write g = Dαf.
In general, a weak derivative may not exist. In particular, Example 2 shows that the Heaviside
function does not admit a weak derivative. Indeed, its distributional derivative is a Dirac
measure (concentrating a unit mass at the origin), not a locally integrable function. On the
other hand, if a weak derivative does exist, then it is unique (up to a set of measure zero).
Lemma 1.1 (uniqueness of weak derivatives). Assume f ∈ L1
loc(Ω) and let g, ˜g ∈ L1
loc(Ω)
be weak α-th derivatives of f, so that
f Dα
φ dx = (−1)|α|
g φ dx = (−1)|α|
˜g φ dx
for all test functions φ ∈ C∞
c (Ω). Then g(x) = ˜g(x) for a.e. x ∈ Ω.
Proof. By the assumptions, the function (g − ˜g) ∈ L1
loc(Ω) satisfies
(g − ˜g)φ dx = 0 for all test functions φ ∈ C∞
c (Ω).
By Corollary A.1 in the Appendix, we thus have g(x) − ˜g(x) = 0 for a.e. x ∈ Ω.
If a function f is twice continuously differentiable, a basic theorem of Calculus states that partial
derivatives commute: fxjxk
= fxkxj . This property remains valid for weak derivatives. To
state this result in full generality, we recall that the sum of two multi-indices α = (α1, . . . , αn)
and β = (β1, . . . , βn) is defined as α + β = (α1 + β1, . . . , αn + βn).
Lemma 1.2 (weak derivatives commute). Assume that f ∈ L1
loc(Ω) has weak derivatives
Dαf for every |α| ≤ k. Then, for every pair of multi-indices α, β with |α| + |β| ≤ k one has
Dα
(Dβ
f) = Dβ
(Dα
f) = Dα+β
f . (1.10)
Proof. Consider any test function φ ∈ C∞
c (Ω). Using the fact that Dβφ ∈ C∞
c (Ω) is a test
function as well, we obtain
Ω
Dα
f Dβ
φ dx = (−1)|α|
Ω f (Dα+βφ) dx
= (−1)|α|
(−1)|α+β|
Ω
(Dα+β
f) φ dx
= (−1)|β|
Ω
(Dα+β
f) φ dx .
6
By definition, this means that Dα+βf = Dβ(Dαf). Exchanging the roles of the multi-indices
α and β in the previous computation one obtains Dα+βf = Dα(Dβf), completing the proof.
The next lemma extends another familiar result, stating that the weak derivative of a limit
coincides with the limit of the weak derivatives.
Lemma 1.3 (convergence of weak derivatives). Consider a sequence of functions fn ∈
L1
loc(Ω). For a fixed multi-index α, assume that each fn admits the weak derivative gn = Dαfn.
If fn → f and gn → g in L1
loc(Ω), then g = Dαf.
Proof. For every test function φ ∈ C∞
c (Ω), a direct computation yields
Ω
g φ dx = lim
n→∞ Ω
gn φ dx = lim
n→∞
(−1)|α|
Ω
fn Dα
φ dx = (−1)|α|
Ω
f Dα
φ dx .
By definition, this means that g is the α-th weak derivative of f.
2 Mollifications
As usual, let Ω ⊆ IRn be an open set. For a given ε > 0, define the open subset
Ωε
.
= {x ∈ IRn
; B(x, ε) ⊂ Ω} . (2.1)
Then for every u ∈ L1
loc(Ω) the mollification
uε(x)
.
= (Jε ∗ u)(x) =
B(x,ε)
Jε(x − y)u(y) dy
is well defined for every x ∈ Ωε. Moreover, uε ∈ C∞(Ωε). A very useful property of the
mollification operator is that it commutes with weak differentiation.
Lemma 2.1 (mollifications). Let Ωε ⊂ Ω be as in (2.1). Assume that a function u ∈
L1
loc(Ω) admits a weak derivative Dαu, for some multi-index α. Then the derivative of the
mollification (which exists in the classical sense) coincides with the mollification of the weak
derivative:
Dα
(Jε ∗ u) = Jε ∗ Dα
u for all x ∈ Ωε . (2.2)
Proof. Observe that, for each fixed x ∈ Ωε, the function φ(y)
.
= Jε(x−y) is in C∞
c (Ω). Hence
we can apply the definition of weak derivative Dαu, using φ as a test function. Writing Dα
x
7
and Dα
y to distinguish differentiation w.r.t. the variables x or y, we thus obtain
Dαuε(x) = Dα
x
Ω
Jε(x − y) u(y) dy
=
Ω
Dα
x Jε(x − y) u(y) dy
= (−1)|α|
Ω
Dα
y Jε(x − y) u(y) dy
= (−1)|α|+|α|
Ω
Jε(x − y) Dα
y u(y) dy
= Jε ∗ Dαu (x).
1/j
Ω
x
Ω
ε
V
Ω
ε
j
Figure 2: Left: the open subset Ωε ⊂ Ω of points having distance > ε from the boundary. Right: the
domain Ω can be covered by countably many open subdomains Vj = Ω1/(j−1) \ Ω1/(j+1).
This property of mollifications stated in Lemma2.1 provides the key tool to relate weak derivatives
with partial derivatives in the classical sense. As a first application, we prove
Corollary 2.1 (constant functions). Let Ω ⊆ IRn be an open, connected set, and assume
u ∈ L1
loc(Ω). If the first order weak derivatives of u satisfy
Dxi u(x) = 0 for i = 1, 2, . . . , n and a.e. x ∈ Ω ,
then u coincides a.e. with a constant function.
Proof. 1. For ε > 0, consider the mollified function uε = Jε ∗ u. By the previous analysis,
uε : Ωε → IR is a smooth function, whose derivatives Dxi uε vanish identically on Ωε. Therefore,
uε must be constant on each connected component of Ωε.
2. Now consider any two points x, y ∈ Ω. Since the open set Ω is connected, there exists
a polygonal path Γ joining x with y and remaining inside Ω. Let δ
.
= minz∈Γ d(z, ∂Ω) be
8
the minimum distance of points in Γ to the boundary of Ω. Then for every ε < δ the whole
polygonal curve Γ is in Ωε. Hence x, y lie in the same connected component of Ωε. In
particular, uε(x) = uε(y).
3. Call ˜u(x)
.
= limε→0 uε(x). By the previous step, ˜u is a constant function on Ω. Moreover,
˜u(x) = u(x) for every Lebesgue point of u, hence almost everywhere on Ω. This concludes the
proof.
Γ
εΩ
x
y
ΩΩ
Figure 3: Left: even if Ω is connected, the subdomain Ωε
.
= {x ∈ Ω ; B(x, ε) ⊆ Ω} may not be
connected. Right: any two points x, y ∈ Ω can be connected by a polygonal path Γ remaining inside
Ω. Hence, if ε > 0 is sufficiently small, x and y belong to the same connected component of Ωε.
In the one-dimensional case, relying again on Lemma 2.1, we now characterize the set of
functions having a weak derivative in L1.
Corollary 2.2 (absolutely continuous functions). Consider an open interval ]a, b[ and
assume that u ∈ L1
loc(]a, b[) has a weak derivative v ∈ L1(]a, b[). Then there exists an absolutely
continuous function ˜u such that
˜u(x) = u(x) for a.e. x ∈ ]a, b[ , (2.3)
v(x) = lim
h→0
˜u(x + h) − ˜u(x)
h
for a.e. x ∈ ]a, b[ . (2.4)
Proof. Let x0 ∈ ]a, b[ be a Lebesgue point of u, and define
˜u(x)
.
= u(x0) +
x
x0
v(y) dy
Clearly ˜u is absolutely continuous and satisfies (2.4).
In order to prove (2.3), let Jε be the standard mollifier and call uε
.
= Jε ∗ u, vε
.
= Jε ∗ v. Then
uε, vε ∈ C∞(]a + ε , b − ε[). Moreover, Lemma 2.1 yields
uε(x) = uε(x0) +
x
x0
vε(y) dy for all x ∈ ]a + ε , b − ε[ . (2.5)
Letting ε → 0 we have uε(x0) → u(x0) because x0 is a Lebesgue point. Moreover, the right
hand side of (2.5) converges to ˜u(x) for every x ∈ ]a, b[ , while the left hand side converges
to u(x) for every Lebesgue point of u (and hence almost everywhere). Therefore (2.3) holds.
9
If f, g ∈ L1
loc(Ω) are weakly differentiable functions, for any constants a, b ∈ IR it is clear that
the linear combination af + bg is also weakly differentiable. Indeed, it satisfies
Dxi (af + bg) = a Dxi f + b Dxi g . (2.6)
We now consider products and compositions of weakly differentiable functions. One should be
aware that, in general, the product of two functions f, g ∈ L1
loc may not be locally integrable.
Similarly, the product of two weakly differentiable functions on IRn may not be weakly differentiable
(see problem 20). For this reason, in the next lemma we shall assume that one of the
two functions is continuously differentiable with uniformly bounded derivatives.
Given two multi-indices α = (α1, . . . , αn) and β = (β1, . . . , βn), we recall that the notation
β ≤ α means βi ≤ αi for every i = 1, . . . , n. Moreover,
α
β
.
=
α!
β! (α − β)!
.
=
α1!
β1! (α1 − β1)!
·
α2!
β2! (α2 − β2)!
· · ·
α2!
β2! (α2 − β2)!
.
Lemma 2.2 (products and compositions of weakly differentiable functions). Let
Ω ⊆ IRn be any open set and consider a function u ∈ L1
loc(Ω) having weak derivatives Dαu of
every order |α| ≤ k.
(i) If η ∈ Ck(Ω), then the product ηu admits weak derivatives up to order k. These are given
by the Leibniz formula
Dα
(ηu) =
β≤α
α
β
Dβ
η Dα−β
u . (2.7)
(ii) Let Ω ⊆ IRn be an open set and let ϕ : Ω → Ω be a Ck bijection whose Jacobian matrix
has a uniformly bounded inverse. Then the composition u ◦ ϕ is a function in L1
loc(Ω ) which
admits weak derivatives up to order k.
Proof. To prove (i), let Jε be the standard mollifier and set uε
.
= Jε ∗ u. Since the Leibniz
formula holds for the product of smooth functions, for every ε > 0 we obtain
Dα
(ηuε) =
β≤α
α
β
Dβ
η Dα−β
uε . (2.8)
For every test function φ ∈ C∞
c (Ω), we thus have
(−1)|α|
Ω
(ηuε)Dα
φ dx =
Ω
Dα
(ηuε)φ dx =
β≤α
α
β Ω
(Dβ
η Dα−β
uε) φ dx .
Notice that, if ε > 0 is small enough so that Supp(φ) ⊂ Ωε, then the above integrals are well
defined. Letting ε → 0 we obtain
(−1)|α|
Ω
(ηu)Dα
φ dx =
Ω


β≤α
α
β
Dβ
η Dα−β
u

 φ dx .
By definition of weak derivative, (2.7) holds.
10
2. We prove (ii) by induction on k. Call y the variable in Ω and x = ϕ(y) the variable in
Ω. By assumption, the n × n Jacobian matrix ∂ϕi
∂yj
)i,j=1,...,n has a uniformly bounded inverse.
Hence the composition u ◦ ϕ lies in L1
loc(Ω ), proving the theorem in the case k = 0.
Next, assume that the result is true for all weak derivatives of order |α| ≤ k − 1. Consider
any test function φ ∈ C∞
c (Ω ). and define the mollification uε
.
= Jε ∗ u. For any ε > 0 small
enough so that ϕ Supp(φ) ⊂ Ωε, we have
−
Ω
(uε ◦ ϕ) · Dyi φ dy =
Ω
Dyi (uε ◦ ϕ) · φ dy
=
Ω


n
j=1
Dxj uε(ϕ(y)) · Dyi ϕj(y)

 · φ(y) dy .
Letting ε → 0 we conclude that the composition u ◦ ϕ admits a first order weak derivative,
given by
Dyi (u ◦ ϕ)(y) =
n
j=1
Dxj u(ϕ(y)) · Dyi ϕj(y). (2.9)
By the inductive assumption, each function Dxj (u ◦ ϕ) admits weak derivatives up to order
k − 1, while Dyi ϕj ∈ Ck−1(Ω ). By part (i) of the theorem, all the products on the right hand
side of (2.9) have weak derivatives up to order k − 1. Using Lemma 1.2 we conclude that the
composition u ◦ ϕ admits weak derivatives up to order k. By induction, this concludes the
proof.
3 Sobolev spaces
Consider an open set Ω ⊆ IRn, fix p ∈ [1, ∞] and let k be a non-negative integer. We say that
an open set Ω is compactly contained in Ω if the closure Ω is a compact subset of Ω.
Definition 3.1 (i) The Sobolev space Wk,p(Ω) is the space of all locally summable functions
u : Ω → IR such that, for every multi-index α with |α| ≤ k, the weak derivative Dαu exists
and belongs to Lp(Ω).
On Wk,p we shall use the norm
u Wk,p
.
=


|α|≤k
Ω
|Dα
u|p
dx


1/p
if 1 ≤ p < ∞, (3.1)
u Wk,∞
.
=
|α|≤k
ess- sup
x∈Ω
|Dα
u| if p = ∞ . (3.2)
(ii) The subspace Wk,p
0 (Ω) ⊆ Wk,p(Ω) is defined as the closure of C∞
c (Ω) in Wk,p(Ω). More
precisely, u ∈ Wk,p
0 (Ω) if and only if there exists a sequence of functions un ∈ C∞
c (Ω) such
that
u − un Wk,p → 0.
11
(iii) By Wk,p
loc (Ω) we mean the space of functions which are locally in Wk,p. These are the
functions u : Ω → IR satisfying the following property. If Ω is an open set compactly contained
in Ω, then the restriction of u to Ω is in Wk,p(Ω ).
Intuitively, one can think of the closed subspace W1,p
0 (Ω) as the space of all functions u ∈
W1,p(Ω) which vanish along the boundary of Ω. More generally, Wk,p
0 (Ω) is a space of functions
whose derivatives Dαu vanish along ∂Ω, for |α| ≤ k − 1.
Definition 3.2 In the special case where p = 2, we define the Hilbert-Sobolev space
Hk(Ω)
.
= Wk,2(Ω). The space Hk(Ω) is endowed with the inner product
u, v Hk
.
=
|α|≤k
Ω
Dα
u Dα
v dx . (3.3)
Similarly, we define Hk
0 (Ω)
.
= Wk,2
0 (Ω).
Theorem 3.1 (basic properties of Sobolev spaces).
(i) Each Sobolev space Wk,p(Ω) is a Banach space.
(ii) The space Wk,p
0 (Ω) is a closed subspace of Wk,p(Ω). Hence it is a Banach space, with the
same norm.
(iii) The spaces Hk(Ω) and Hk
0 (Ω) are Hilbert spaces.
Proof. 1. Let u, v ∈ Wk,p(Ω). For |α| ≤ k, call Dαu, Dαv their weak derivatives. Then,
for any λ, µ ∈ IR, the linear combination λu + µv is a locally integrable function. One easily
checks that its weak derivatives are
Dα
(λu + µv) = λDα
u + µDα
v . (3.4)
Therefore, Dα(λu + µv) ∈ Lp(Ω) for every |α| ≤ k. This proves that Wk,p(Ω) is a vector
space.
2. Next, we check that (3.1) and (3.2) satisfy the conditions (N1)–(N3) defining a norm.
Indeed, for λ ∈ IR and u ∈ Wk,p one has
λu Wk,p = |λ| u Wk,p ,
u Wk,p ≥ u Lp ≥ 0 ,
with equality holding if and only if u = 0.
12
Moreover, if u, v ∈ Wk,p(Ω), then for 1 ≤ p < ∞ Minkowski’s inequality yields
u + v Wk,p =


|α|≤k
Dα
u + Dα
v p
Lp


1/p
≤


|α|≤k
Dα
u Lp + Dα
v Lp
p


1/p
≤


|α|≤k
Dα
u p
Lp


1/p
+


|α|≤k
Dα
v p
Lp


1/p
= u Wk,p + v Wk,p .
In the case p = ∞, the above computation is replaced by
u+v Wk,∞ =
|α|≤k
Dα
u+Dα
v L∞ ≤
|α|≤k
Dα
u L∞ + Dα
v L∞ = u Wk,∞ + v Wk,∞ .
3. To conclude the proof of (i), we need to show that the space Wk,p(Ω) is complete, hence it
is a Banach space.
Let (un)n≥1 be a Cauchy sequence in Wk,p(Ω). For any multi-index α with |α| ≤ k, the
sequence of weak derivatives Dαun is Cauchy in Lp(Ω). Since the space Lp(Ω) is complete,
there exist functions u and uα, such that
un − u Lp → 0, Dα
un − uα Lp → 0 for all |α| ≤ k . (3.5)
By Lemma 1.3, the limit function uα is precisely the weak derivative Dαu. Since this holds for
every multi-index α with |α| ≤ k, the convergence un → u holds in Wk,p(Ω). This completes
the proof of (i).
4. The fact that Wk,p
0 (Ω) is a closed subspace of Wk,p(Ω) follows immediately from the
definition. The fact that (3.3) is an inner product is also clear.
Example 7. Let Ω = ]a, b[ be an open interval. By Corollary 2.2, each element of the
space W1,p(]a, b[) coincides a.e. with an absolutely continuous function f : ]a, b[→ IR having
derivative f ∈ Lp(]a, b[).
Example 8. Let Ω = B(0, 1) ⊂ IRn be the open ball centered at the origin with radius one.
Fix γ > 0 and consider the radially symmetric function
u(x)
.
= |x|−γ
=
n
i=1
x2
i
−γ/2
0 < |x| < 1.
13
−γ
x2
x1
u(x) = |x|
Figure 4: For certain values of p, n a function u ∈ W1,p
(IRn
) may not be continuous, or bounded.
Observe that u ∈ C1(Ω \ {0}). Outside the origin, its partial derivatives are computed as
uxi = −
γ
2
n
i=1
x2
i
−(γ/2)−1
2xi =
−γxi
|x|γ+2
. (3.6)
Hence, the gradient u = (ux1 , . . . , uxn ) has norm
| u(x)| =
n
i=1
|uxi (x)|2
1/2
=
γ
|x|γ+1
.
On the open set Ω\{0}, the function u clearly admits weak derivatives of all orders, and these
coincide with the classical ones.
We wish to understand in which cases the formula (3.6) defines the weak derivatives of u on
the entire domain Ω. This means
Ω
uxi φ dx = −
Ω
u φxi dx
for every smooth function φ ∈ C∞
c (Ω) whose support is a compact subset of Ω (and not only
for those functions φ whose support is a compact subset of Ω \ {0}).
Observe that, for any ε > 0, one has
ε<|x|<1
(uφ)xi dx =
|x|=ε
u(x)φ(x) νi(x) dS
where dS is the (n − 1)-dimensional measure on the surface of the ball B(0, ε), and νi(x) =
−xi/|x|, so that ν = (ν1, ν2, . . . , νn) is the unit normal pointing toward the interior of the ball
14
B(0, ε). Since φ is a bounded continuous function, we have
|x|=ε
|u(x)φ(x) νi(x)| dS ≤ φ L∞ ε−γ
σnεn−1
→ 0 as ε → 0
provided that n − 1 > γ. In other words, if γ < n − 1, then
Ω
−γxi
|x|γ+2
φ(x) dx =
Ω
1
|x|γ
φxi (x) dx for every test function φ ∈ C∞
c (Ω) ,
and the locally integrable function uxi defined at (3.6) is in fact the weak derivative of u on
the whole domain Ω.
Observe that
Ω
1
|x|γ+1
p
dx =
x∈IRn, |x|<1
|x|−p(γ+1)
dx = σn
1
0
rn−1
r−p(γ+1)
dr < ∞
if and only if n − 1 − p(γ + 1) > −1, i.e. γ < n−p
p .
The previous computations show that, if 0 < γ < n−p
p , then u ∈ W1,p(Ω).
Notice that u is absolutely continuous (in fact, smooth) on a.e. line parallel to one of the
coordinate axes. However, there is no way to change u on a set of measure zero, so that it
becomes continuous on the whole domain Ω.
An important question in the theory of Sobolev spaces is whether one can estimate the norm
of a function in terms of the norm of its first derivatives. The following result provides an
elementary estimate in this direction. It is valid for domains Ω which are contained in a slab,
say
Ω ⊆ {x = (x1, x2, . . . , xn) ; a < x1 < b} . (3.7)
Theorem 3.2 (Poincare’s inequality - I). Let Ω ⊂ IRn be an open set which satisfies (3.7)
for some a, b ∈ IR. Then, every u ∈ H1
0 (Ω) satisfies
u L2(Ω) ≤ 2(b − a) Dx1 u L2(Ω) . (3.8)
Proof. 1. Assume first that u ∈ C∞
c (Ω). We extend u to the whole space IRn by setting
u(x) = 0 for x /∈ Ω. Using the variables x = (x1, x ) with x = (x2, . . . , xn), we compute
u2
(x1, x ) =
x1
a
2uux1 (t, x ) dt .
An integration by parts yields
u 2
L2 =
IRn
u2
(x) dx =
IRn−1
b
a
1 ·
x1
a
2uux1 (x1, x ) dt dx1 dx
=
IRn−1
b
a
(b − x1) 2uux1 (t, x ) dx1 dx ≤ 2(b − a)
IRn
|u| |ux1 | dx
≤ 2(b − a) u L2 ux1 L2 .
15
Dividing both sides by u L2 we obtain (3.8), for every u ∈ C∞
c (Ω).
2. Now consider any u ∈ H1
0 (Ω). By assumption there exists a sequence of functions un ∈
C∞
c (Ω) such that un − u H1 → 0. By the previous step, this implies
u L2 = lim
n→∞
un L2 ≤ lim
n→∞
2(b − a) Dx1 un L2 = 2(b − a) Dx1 u L2 .
To proceed in the analysis of Sobolev spaces, we need to derive some more properties of weak
derivatives.
Theorem 3.3 (properties of weak derivatives). Let Ω ⊆ IRn be an open set, p ∈ [1, ∞],
and |α| ≤ k. If u, v ∈ Wk,p(Ω), then
(i) The restriction of u to any open set Ω ⊂ Ω is in the space Wk,p(Ω).
(ii) Dαu ∈ Wk−|α|,p(Ω).
(iii) If η ∈ Ck(Ω), then the product satisfies η u ∈ Wk,p(Ω). Moreover there exists a constant
C depending on Ω and on η Ck but not on u, such that
ηu Wk,p(Ω) ≤ C u Wk,p(Ω) . (3.9)
(iv) Let Ω ⊆ IRn be an open set and let ϕ : Ω → Ω be a Ck diffeomorphism whose Jacobian
matrix has a uniformly bounded inverse. Then the composition satisfies u ◦ ϕ ∈ Wk,p(Ω ).
Moreover there exists a constant C, depending on Ω and on ϕ Ck but not on u, such that
u ◦ ϕ Wk,p(Ω ) ≤ C u Wk,p(Ω) . (3.10)
Proof. The statement (i) is an obvious consequence of the definitions, while (ii) follows from
Lemma 1.2.
To prove (iii), by assumption all derivatives of η are bounded, namely
Dβ
η L∞ ≤ η Ck for all |β| ≤ k .
Hence the bound (3.9) follows from the representation formula (2.7).
We prove (iv) by induction on k. By assumption, the n × n Jacobian matrix (Dxi ϕj)i,j=1,...,n
has a uniformly bounded inverse. Hence the case k = 0 is clear.
Next, assume that the result is true for k = 0, 1, . . . , m − 1. If u ∈ Wm,p(Ω), we have
Dxi (u ◦ ϕ) Wm−1,p(Ω ) ≤ C u Wm−1,p(Ω) ϕ Cm(Ω ) ≤ C u Wm,p(Ω) ,
showing that the result is true also for k = m. By induction, this achieves the proof.
16
4 Approximations and extensions of Sobolev functions
Theorem 4.1 (approximation with smooth functions). Let Ω ⊆ IRn be an open set.
Let u ∈ Wk,p(Ω) with 1 ≤ p < ∞. Then there exists a sequence of functions uk ∈ C∞(Ω) such
that uk − u Wk,p(Ω) → 0 as n → ∞.
Proof. 1. Let ε > 0 be given. Consider the following locally finite open covering of the set Ω,
shown in fig. 2:
V1
.
= x ∈ Ω ; d(x, ∂Ω) >
1
2
, Vj
.
= x ∈ Ω ;
1
j + 1
< d(x, ∂Ω) <
1
j − 1
j = 2, 3, . . .
Let η1, η2, . . . be a smooth partition of unity subordinate to the above covering. By Theorem
7.3, for every j ≥ 0 the product ηju is in Wk,p(Ω). By construction, it has support
contained in Vj.
2. Consider the mollifications J ∗ (ηju). By Lemma 7.3, for every |α| ≤ k we have
Dα
(J ∗ (ηju)) = J ∗ (Dα
(ηju)) → Dα
(ηju)
as → 0. Since each ηj has compact support, here the convergence takes place in Lp(Ω).
Therefore, for each j ≥ 0 we can find 0 < j < 2−j small enough so that
ηju − J j ∗ (ηju) Wk,p(Ω) ≤ ε 2−j
.
3. Consider the function
U
.
=
∞
j=1
Jεj ∗ (ηju)
Notice that the above series may not converge in Wk,p. However, it is certainly pointwise
convergent because every compact set K ⊂ Ω intersects finitely many of the sets Vj. Restricted
to K, the above sum contains only finitely many non-zero terms. Since each term is smooth,
this implies U ∈ C∞(Ω).
4. Consider the subdomains
Ω1/n
.
= x ∈ Ω ; d(x, ∂Ω) >
1
n
.
Recalling that j ηj(x) ≡ 1, for every n ≥ 1 we find
U − u Wk,p(Ω1/n) ≤
n+2
j=1
ηju − Jεj ∗ (ηju) Wk,p(Ω1/n) ≤
n+2
j=1
ε 2−j
≤ ε.
This yields
U − u Wk,p(Ω) = sup
n≥1
U − u Wk,p(Ω1/n) ≤ ε .
Since ε > 0 was arbitrary, this proves that the set of C∞ function is dense on Wk,p(Ω).
17
Using the above approximation result, we obtain a first regularity theorem for Sobolev functions
(see fig. 5).
Theorem 4.2 (relation between weak and strong derivatives). Let u ∈ W1,1(Ω),
where Ω ⊆ IRn is an open set having the form
Ω = x = (x, x ) ; x
.
= (x2, . . . , xn) ∈ Ω , α(x ) < x1 < β(x ) (4.1)
(possibly with α ≡ −∞ or β ≡ +∞). Then there exists a function ˜u with ˜u(x) = u(x) for
a.e. x ∈ Ω, such that the following holds. For a.e. x = (x2, . . . , xn) ∈ Ω ⊂ IRn−1 (w.r.t. the
(n − 1)-dimensional Lebesgue measure), the map
x1 → ˜u(x1, x )
is absolutely continuous. Its derivative coincides a.e. with the weak derivative Dx1 u.
x1
Ω
x =1 α(x , x )2 3 x =x
x2
3
1
β(x , x )2 3
’Ω
Figure 5: The domain Ω at (4.1). If u has a weak derivative Dx1
u ∈ L1
(Ω), then (by possibly changing
its values on a set of measure zero), the function u is absolutely continuous on almost every segment
parallel to the x1-axis, and its partial derivative ∂u/∂x1 coincides a.e. with the weak derivative.
Proof. 1. By the previous theorem, there exists a sequence of functions uk ∈ C∞(Ω) such
that
uk − u W1,1 < 2−k
. (4.2)
We claim that this implies the pointwise convergence
uk(x) → u(x), Dx1 uk(x) → Dx1 u(x) for a.e. x ∈ Ω .
Indeed, consider the functions
f(x)
.
= |u1(x)|+
∞
k=1
|uk+1(x)−uk(x)| , g(x)
.
= |Dx1 u1(x)|+
∞
k=1
Dx1 uk+1(x)−Dx1 uk(x) .
(4.3)
By (4.2), there holds
uk − uk+1 W1,1 ≤ 21−k
,
18
hence f, g ∈ L1(Ω) and the series in (4.3) are absolutely convergent for a.e. x ∈ Ω. Therefore,
they converge pointwise almost everywhere. Moreover, we have the bounds
|uk(x)| ≤ f(x) , |Dx1 uk(x)| ≤ g(x) for all n ≥ 1, x ∈ Ω . (4.4)
2. Since f, g ∈ L1(Ω), by Fubini’s theorem there exists a null set N ⊂ Ω (w.r.t. the n − 1
dimensional Lebesgue measure) such that, for every x ∈ Ω \ N one has
β(x )
α(x )
f(x1, x ) dx1 < ∞ ,
β(x )
α(x )
g(x1, x ) dx1 < ∞. (4.5)
Fix such a point x ∈ Ω \ N. Choose a point y1 ∈ ]α(x ) , β(x )[ where the pointwise
convergence uk(y1, x ) → u(y1, x ) holds. For every α(x ) < x1 < β(x ), since uk is smooth we
have
uk(x1, x ) = uk(y1, x ) +
x1
y1
Dx1 uk(s, x ) ds . (4.6)
We now let n → ∞ in (4.6). By (4.4) and (4.5), the functions Dx1 uk(·, x ) are all bounded by
the integrable function g(·, x ) ∈ L1. By the dominated convergence theorem, the right hand
side of (4.6) thus converges to
˜u(x1, x )
.
= u(y1, x ) +
x1
y1
Dx1 u(s, x ) ds . (4.7)
Clearly, the right hand side of (4.7) is an absolutely continuous function of the scalar variable
x1. On the other hand, the left hand side satisfies
˜u(x1, x )
.
= lim
n→∞
uk(x1, x ) = u(x1, x ) for a.e. x1 ∈ [α(x ), β(x )].
This achieves the proof.
Remark 4.1 (i) It is clear that a similar result holds for any other derivative Dxi u, with
i = 1, 2, . . . , n.
(ii) If u ∈ Wk,p(Ω) and Ω ⊂ Ω, then the restriction of u to Ω lies in the space Wk,p(Ω). Even
if the open set Ω has a complicated topology, the result of Theorem 4.2 can be applied to any
cylindrical subdomain Ω ⊂ Ω admitting the representation (4.1).
(iii) If Ω ⊂ IRn is a bounded open set and u ∈ Wk,p(Ω), then u ∈ Wk,q(Ω) for every q ∈ [1, p].
The next theorem shows that, given a bounded open domain Ω ⊂ IRn with C1 boundary, each
function u ∈ W1,p(Ω) can be extended to a function Eu ∈ W1,p(IRn).
Theorem 4.3 (extension operators). Let Ω ⊂⊂ Ω ⊂ IRn be open sets, such that the
closure of Ω is a compact subset of Ω. Moreover, assume that ∂Ω ∈ C1. Then there exists a
bounded linear operator E : W1,p(Ω) → W1,p(IRn) and a constant C such that
(i) Eu(x) = u(x) for a.e. x ∈ Ω,
(ii) Eu(x) = 0 for x /∈ Ω,
(iii) One has the bound Eu W1,p(IRn) ≤ C u W1,p(Ω).
19
Proof. 1. We first prove that the same result holds in the case where the domain is a
half space: Ω = {x = (x1, x2, . . . , xn) ; x1 > 0}, and Ω = IRn. In this case, any function
u ∈ W1,p(Ω) can be extended to the whole space IRn by reflection, i.e. by setting
(E u)(x1, x2, . . . , xn)
.
= u ( |x1|, x2, x3, . . . , xn) . (4.8)
By Theorem 4.2, for every i ∈ {1, . . . , n} the function u is absolutely continuous along a.e. line
parallel to the xi-axis. Hence the same is true of the extension E u. A straightforward
computation involving integration by parts shows that the first order weak derivatives of E u
exist on the entire space IRn and satisfy
Dx1 E u(−x1, x2, . . . , xn) = − Dx1 u(x1, x2, . . . , xn),
Dxj E u(−x1, x2, . . . , xn) = Dxj u(x1, x2, . . . , xn) (j = 2, . . . , n),
for all x1 > 0, x2, . . . , xn ∈ IR. The extension operator E : W1,p(Ω) → W1,p(IRn) defined at
(4.8) is clearly linear and bounded, because
E u W1,p(IRn) ≤ 2 u W1,p(Ω) .
0
xj
Ω
~
Ω
1
y
xi
Bi
B
2
’y = (y , ... , y )n
i
ϕ
B
+
Figure 6: The open covering of the set Ω. For every ball Bi = B(xi, ri) there is a C1
bijection ϕi
mapping the open unit ball B ⊂ IRn
onto Bi. For those balls Bi having center on the boundary Ω, the
intersection Bi ∩ Ω is mapped into B+
= B ∩ {y1 > 0}.
2. To handle the general case, we use a partition of unity. For every x ∈ Ω (the closure of Ω),
choose a radius rx > 0 such that the open ball B(x, rx) centered at x with radius rx0 satisfies
the following conditions
- If x ∈ Ω, then B(x, rx) ⊂ Ω.
- If x ∈ ∂Ω, then B(x, rx) ⊂ Ω. Moreover, calling B
.
= B(0, 1) the open unit ball in IRn, there
exists a C1 bijection ϕx : B → B(x, rx), whose inverse is also C1, which maps the half ball
B+ .
= y = (y1, y2, . . . , yn) ;
n
i=1
y2
i < 1 , y1 > 0
onto the set B(x, rx) ∩ Ω.
20
Choosing rx > 0 sufficiently small, the existence of such a bijection follows from the assumption
that Ω has a C1 boundary.
Since Ω is bounded, its closure Ω is compact. Hence it can be covered with finitely many balls
Bi = B(xi, ri), i = 1, . . . , N. Let ϕi : B → Bi be the corresponding bijections. Recall that ϕi
maps B+ onto Bi ∩ Ω.
3. Let η1, . . . , ηN be a smooth partition of unity subordinated to the above covering. For
every x ∈ Ω we thus have
u(x) =
N
i=1
ηi(x)u(x) . (4.9)
We split the set of indices as
{1, 2, . . . , N} = I ∪ J ,
where I contains the indices with xi ∈ Ω while J contains the indices with xi ∈ ∂Ω.
For every i ∈ J , we have ηiu ∈ W1,p(Bi ∩Ω). Hence by Theorem 3.3 (iv), one has (ηiu)◦ϕi ∈
W1,p(B+). Applying the extension operator E defined at (4.8) one obtains
E (ηiu) ◦ ϕi ∈ W1,p
(B+
) , E (ηiu) ◦ ϕi ◦ ϕ−1
i ∈ W1,p
(Bi) .
Summing together all these extensions, we define
Eu
.
=
i∈I
ηiu +
i∈J
E (ηiu) ◦ ϕi ◦ ϕ−1
i .
It is now clear that the extension operator E satisfies all requirements. Indeed, (i) follows
from (4.9). The property (ii) stems from the fact that, for every u ∈ W1,p(Ω), the support
of Eu is contained in ∪N
i=1Bi ⊆ Ω. Finally, since E is defined as the sum of finitely many
bounded linear operators, the bound (iii) holds, for some constant C.
For any open set Ω ⊆ IRn, in Theorem 4.1 we approximated a function u ∈ Wk,p(Ω) with C∞
functions uk defined on the same open set Ω. In principle, these approximating functions uk
might have wild behavior near the boundary of Ω. Assuming that the boundary ∂Ω is C1,
using the extension theorem, we now show that u can be approximated in W1,p by smooth
functions which are defined on the entire space IRn.
Corollary 4.1 Let Ω ⊂ IRn be a bounded open set with C1 boundary. Given any u ∈ W1,p(Ω)
with 1 ≤ p < ∞, there exists a sequence of smooth functions uk ∈ C∞
c (IRn) such that the
restrictions of uk to Ω satisfy
lim
k→∞
uk − u W1,p(Ω) = 0 . (4.10)
Moreover,
uk W1,p(IRn) ≤ C u W1,p(Ω) , (4.11)
for some constant C depending on p and Ω but not on u.
21
Proof. Let Ω = B(Ω, 1) be the open neighborhood of radius one around the set Ω. By
Theorem 4.3 the function u admits an extension Eu ∈ W1,p(IRn) which vanishes Ω. Then the
mollifications uk = J1/k ∗ Eu ∈ C∞
c (IRn) satisfy all requirements (4.10)-(4.11). Indeed,
uk W1,p(IRn) ≤ Eu W1,p(IRn) ≤ C u W1,p(Ω) , (4.12)
lim
k→∞
uk − u W1,p(Ω) ≤ lim
k→∞
uk − Eu W1,p(IRn) = 0.
Γ
Ω
x
1
x2
u
Ω
1
x
uk
x2
Figure 7: Making sense of the trace of a function u ∈ W1,p
(Ω). Left: the thick curve Γ is a portion of
the boundary which admits the representation {x1 = α(x2)}. Since u is absolutely continuous along
almost all lines parallel to the x1-axis, along Γ one can define the trace of of u by taking pointwise
limits along horizontal lines. Right: the standard way to construct a trace. Given a sequence of smooth
functions uk ∈ C∞
(IRn
) whose restrictions uk Ω
converge to u in W1,p
(Ω), one defines the trace as
Tu
.
= limk→∞ uk ∂Ω
. This limit is well defined in L1
(∂Ω) and does not depend on approximating
sequence.
5 Traces
Let Ω ⊂ IRn be a bounded open set with C1 boundary. Call Lp(∂Ω) the corresponding space
of functions v : ∂Ω → IR, with norm
v Lp(∂Ω)
.
=
∂Ω
|v|p
dS
1/p
(5.1)
where dS denotes the (n − 1)-dimensional surface measure on ∂Ω.
Given a function u ∈ W1,p(Ω), we seek to define the “boundary values” of u. Before giving a
precise theorem in this direction, let us consider various possible approaches to this problem.
(i) If u is uniformly continuous on the open set Ω, one can simply extend u by continuity
to the closure Ω. For any boundary point x ∈ ∂Ω, we can thus define the Trace of u at
x by setting
Tu(x)
.
= lim
y→x , y∈Ω
u(y) . (5.2)
By uniform continuity, this limit is well defined and yields a continuous function Tu :
∂Ω → IR. In general, however, a function u ∈ W1,p(Ω) may well be discontinuous.
Hence the limit in (5.2) may not exist.
22
(ii) Consider a portion of the boundary ∂Ω that can be written in the form (see fig. 7, left)
{x = (x1, . . . , xn)
.
= (x1, x ) ; x1 = α(x )}.
By Theorem 4.2, the function u is absolutely continuous along a.e. line parallel to the
x1-axis. Therefore the pointwise limit
v(α(x ), x ) = lim
x1→α(x ), (x1,x )∈Ω
u(x1, x )
exists for a.e. x .
This provides is a good way to think about the trace of a function. With this approach,
however, the properties of these pointwise limits are not easy to derive. Also, it is not
immediately clear whether by taking the limit along the x1 direction or along some other
direction one always obtains the same boundary values.
(iii) The standard approach, which we shall adopt in the sequel, relies on the approximation of
u with smooth functions uk defined on the entire space IRn (see fig. 7). By Corollary 2.2
there exists a sequence of smooth functions uk ∈ C∞(IRn) such that their restrictions
to Ω approach u in W1,p(Ω). A more careful analysis shows that the restrictions of
these smooth functions to ∂Ω form a Cauchy sequence in Lp(∂Ω). Hence it determines a
unique limit v = Tu ∈ Lp(∂Ω). This limit, that does not depend on the approximating
sequence (uk)k≥1 is called the trace of u on ∂Ω.
Theorem 5.1 (trace operator). Let Ω ⊂ IRn be a bounded open set with C1 boundary, and
let 1 ≤ p < ∞. Then there exists a bounded linear operator T : W1,p(Ω) → Lp(∂Ω) with the
following properties.
(i) If u ∈ W1,p(Ω) ∩ C0(Ω), then
Tu = u|∂Ω . (5.3)
(ii) There exists a constant C depending only on p and on the set Ω such that
Tu Lp(∂Ω) ≤ C u W1,p(Ω) . (5.4)
Proof. Given any function u ∈ C∞(IRn), we claim that its restrictions to Ω and to ∂Ω satisfy
u Lp(∂Ω) ≤ C u W1,p(Ω) , (5.5)
for some constant C depending only on p and on the set Ω. This key inequality will be proved
in the next two steps.
1. Let u ∈ C∞(IRn) be a smooth function which vanishes outside the unit ball B. Consider the
(n − 1)-dimensional disc Γ = {(x1, . . . , xn) ; i x2
i < 1 , x1 = 0 , }. Setting x = (x2, . . . , xn)
and B+ .
= B ∩ {x1 > 0}, we claim that
Γ
|u(x )|p
dx ≤ C1
B+
|u(x)|p
+ | u(x)|p
dx (5.6)
23
for some constant C1 depending only on p. Indeed, since u vanishes outside B, using Young’s
inequality ab ≤ ap
p + bq
q with q = p
p−1 we find
Γ
|u(0, x )|p
dx = −
Γ
∞
0
d
dx1
|u(x1, x )|p
dx1 dx
≤
Γ
∞
0
p|u|p−1
|ux1 | dx1 dx ≤ C1
B+
|u|p
+ |ux1 |p
dx .
2. As in the proof of Theorem 4.3 (see fig. 6), we can cover the compact set Ω with finitely
many balls B1, . . . , BN having the following properties. For every i ∈ {1, . . . , N} there exists
a C1 bijection ϕi : B → Bi with C1 inverse, such that either Bi ⊆ Ω, or
ϕi(B+
) = (Bi ∩ Ω)
Let η1, . . . , ηN be a smooth partition of unity subordinated to this covering. Call J ⊆
{1, . . . , N} the set of indices for which ϕi(B) intersects the boundary of Ω. If u ∈ C∞(IRn),
we now have the estimate
u p
Lp(∂Ω) ≤
i
uηi
p
Lp(∂Ω) ≤
i
C2 · (ηiu) ◦ ϕi
p
Lp(Γ)
≤ C3
i∈J B+
|(ηiu) ◦ ϕi|p
+ ((ηiu) ◦ ϕi)
p
dy
≤ C4
i∈J Bi∩Ω
|ηiu|p
+ | (ηiu)|p
dx ≤ C5
Ω
|u|p
+ | u|p
dx .
for suitable constants C2, . . . , C5. Notice that the third inequality is a consequence of (5.6). To
obtain the other inequalities, we used the fact that the functions ηi, ϕi, and ϕ−1
i are bounded
with uniformly bounded derivatives. The above arguments establish the a priori bound (5.5).
3. Next consider any function u ∈ W1,p(Ω). By Corollary 4.1 there exists a sequence of
functions um ∈ C∞(IRn) such that um − u W1,p(Ω) → 0. By the previous step, for any
n, m ≥ 1 we have
lim sup
m,n→∞
um − un Lp(∂Ω) ≤ C lim sup
m,n→∞
um − un W1,p(Ω) = 0. (5.7)
Hence the restrictions uk|∂Ω provide a Cauchy sequence in Lp(∂Ω). The limit of this sequence
is a function Tu ∈ Lp(∂Ω), which we call the Trace of u. By (5.7), this limit does not depend
on the choice of the approximating sequence.
4. It remains to check that the operator u → Tu constructed in the previous step satisfies the
required properties (i)–(ii).
If u ∈ W1,p(Ω) ∩ C0(Ω), then its extension Eu ∈ W1,p(IRn) constructed in Theorem 4.3 is
continuous on the whole space IRn. The sequence of mollifications uk
.
= J1/k ∗ (Eu) converge
to Eu uniformly on compact sets. In particular, they converge to u uniformly on ∂Ω. This
proves (i). By the definition of trace, the bound (ii) is an immediate consequence of (5.5).
24
Remark 5.1 Let u → Eu be an extension operator, constructed in Theorem 4.3. Then the
trace operator can be defined as
Tu = lim
ε→0
(Jε ∗ Eu)
∂Ω
(5.8)
where Jε denotes a mollifier and the limit takes place in Lp(∂Ω).
6 Embedding theorems
In one space dimension, a function u : IR → IR which admits a weak derivative Du ∈ L1(IR)
is absolutely continuous (after changing its values on a set of measure zero). On the other
hand, if Ω ⊆ IRn with n ≥ 2, there exist functions u ∈ W1,p(Ω) which are not continuous, and
not even bounded. This is indeed the case of the function u(x) = |x|−γ, for 0 < γ < n−p
p .
In several applications to PDEs or to the Calculus of Variations, it is important to understand
the degree of regularity enjoyed by functions u ∈ Wk,p(IRn). We shall prove two basic results
in this direction.
1. (Morrey). If p > n, then every function u ∈ W1,p(IRn) is H¨older continuous (after a
modification on a set of measure zero).
2. (Gagliardo-Nirenberg). If p < n, then every function u ∈ W1,p(IRn) lies in the space
Lp∗
(IRn), with the larger exponent p∗ = p + p2
n−p .
In both cases, the result can be stated as an embedding theorem: after a modification on a set
of measure zero, each function u ∈ W1,p(Ω) also lies in some other Banach space X. Typically
X = C0,γ, or X = Lq for some q > p. The basic approach is as follows:
I - Prove an a priori inequality valid for all smooth functions. Given any function
u ∈ C∞ ∩ Wk,p(Ω), one proves that u also lies in another Banach space X, and there exists a
constant C depending on k, p, Ω but not on u, such that
u X ≤ C u Wk,p for all u ∈ C∞
∩ Wk,p
(Ω) . (6.1)
II - Extend the embedding to the entire space, by continuity. Since C∞ is dense
in Wk,p, for every u ∈ Wk,p(Ω) we can find a sequence of functions un ∈ C∞ such that
u − un Wk,p → 0. By (6.1),
lim sup
m,n→∞
um − un X ≤ lim sup
m,n→∞
C um − un Wk,p = 0 .
Therefore the functions un form a Cauchy sequence also in the space X. By completeness,
un → ˜u for some ˜u ∈ X. Observing that ˜u(x) = u(x) for a.e. x ∈ Ω, we conclude that, up to
a modification on a set N ⊂ Ω of measure zero, each function u ∈ Wk,p(Ω) lies also in the
space X.
25
6.1 Morrey’s inequality
In this section we prove that, if u ∈ W1,p(IRn), where the exponent p is higher than the
dimension n of the space, then u coincides a.e. with a H¨older continuous function.
Theorem 6.1 (Morrey’s inequality). Assume n < p < ∞ and set γ
.
= 1 − n
p > 0. Then
there exists a constant C, depending only on p and n, such that
u C0,γ(IRn) ≤ C u W1,p(IRn) (6.2)
for every u ∈ C1(IRn) ∩ W1,p(IRn).
z yy ’ ρ ρx
1
0 0
Γ
x
1
B
1
r
ξ = (ξ , ... , ξ )
2H n
Figure 8: Proving Morrey’s inequality. Left: the values u(y) and u(y ) are compared with the average
value of u on the (n − 1)-dimensional ball centered at the mid-point z. Center and right: a point x in
the cone Γ is described in terms of the coordinates (r, ξ) ∈ [0, ρ] × B1 .
Proof. 0. Before giving the actual proof, we outline the underlying idea. From an integral
estimate on the gradient of the function u, say
IRn
| u(x)|p
dx ≤ C0, (6.3)
we seek a pointwise estimate of the form
|u(y) − u(y )| ≤ C1|y − y |γ
for all y, y ∈ IRn
. (6.4)
To achieve (6.4), a natural attempt is to write
|u(y)−u(y )| =
1
0
d
dθ
u θy + (1 − θ)y dθ ≤
1
0
u(θy+(1−θ)y ) |y−y | dθ . (6.5)
However, the integral on the right hand side of (6.5) only involves values of u over the
segment joining y with y . If the dimension of the space is n > 1, this segment has zero
measure. Hence the integral in (6.5) can be arbitrarily large, even if the integral in (6.3) is
26
small. To address this difficulty, we shall compare both values u(y), u(y ) with the average
value uB of the function u over an (n−1)-dimensional ball centered at the midpoint z = y+y
2 ,
as shown in Figure 8, left. Notice that the difference |u(y) − uB| can be estimated by an
integral of | u| ranging over a cone of dimension n. In this way the bound (6.3) can thus be
brought into play.
1. We now begin the proof, with a preliminary computation. On IRn, consider the cone
Γ
.
=



x = (x1, x2, . . . , xn) ;
n
j=2
x2
j ≤ x2
1 , 0 < x1 < ρ



and the function
ψ(x) =
1
xn−1
1
. (6.6)
Let q = p
p−1 be the conjugate exponent of p, so that 1
p + 1
q = 1. We compute
ψ q
Lq(Γ) =
Γ
1
xn−1
1
q
dx =
ρ
0
cn−1xn−1
1
1
xn−1
1
q
dx1 = cn−1
ρ
0
s(n−1)(1−q)
ds ,
where the constant cn−1 gives the volume of the unit ball in IRn−1. Therefore, ψ ∈ Lq(Γ) if
and only if n < p. In this case,
ψ Lq(Γ) = cn−1
ρ
0
s
−n−1
p−1 ds
1
q
= c ρ
p−n
p−1
p−1
p
= c ρ
p−n
p , (6.7)
for some constant c depending only on n and p.
2. Consider any two distinct points y, y ∈ IRn. Let ρ
.
= 1
2|y − y |. The hyperplane passing
through the midpoint z
.
= y+y
2 and perpendicular to the vector y − y has equation
H = x ∈ IRn
; x − z , y − y = 0 .
Inside H, consider the (n − 1)-dimensional ball centered at z with radius ρ,
Bρ
.
= x ∈ H ; |x − z| < ρ .
Calling uA the average value of u on the ball Bρ, the difference |u(y)−u(y )| will be estimated
as
|u(y) − u(y )| ≤ |u(y) − uA| + uA − u(y ) . (6.8)
3. By a translation and a rotation of coordinates, we can assume
y = (0, . . . , 0) ∈ IRn
, Bρ = x = (x1, x2, . . . , xn) ; x1 = ρ ,
n
i=2
x2
i ≤ ρ2
.
To compute the average value uA, let B1 be the unit ball in IRn−1, and let cn−1 be its (n − 1)dimensional
measure. Points in the cone Γ will be described using an alternative system of
27
coordinates. To (r, ξ) = (r, ξ2, . . . , ξn) ∈ [0, ρ] × B1 we associate the point x(r, ξ) ∈ Γ,
defined by
(x1, x2, . . . , xn) = (r, rξ) = (r, rξ2, . . . , rξn). (6.9)
Define U(r, ξ) = u(r, rξ), and observe that U(0, ξ) = u(0) for every ξ. Therefore
U(ρ, ξ) = U(0, ξ) +
ρ
0
∂
∂r
U(r, ξ) dr ,
uB − u(0) =
1
cn−1 B1
ρ
0
∂
∂r
U(r, ξ) dr dξ . (6.10)
We now change variables, transforming the integral (6.10) over [0, ρ] × B1 into an integral
over the cone Γ. Computing the Jacobian matrix of the transformation (6.9) we find that its
determinant is rn−1, hence
dx1 dx2 · · · dxn = rn−1
dr dξ2 · · · dξn .
Moreover, since |ξ| ≤ 1, the directional derivative of u in the direction of the vector
(1, ξ2, . . . , ξn) is estimated by
∂
∂r
U(r, ξ) = ux1 +
n
i=2
ξiuxi ≤ 2| u(r, ξ)| . (6.11)
Using (6.11) in (6.10), and the estimate (6.7) on the Lq norm of the function ψ(x)
.
= x1−n
1 ,
we obtain
uA − u(0) ≤
2
cn−1 Γ
1
xn−1
1
| u(x)| dx
≤
2
cn−1
ψ Lq(Γ) u Lp(Γ) q =
p
p − 1
≤ C ρ
p−n
p u W1,p(IRn)
(6.12)
for some constant C. Notice that the last two steps follow from H¨older’s inequality and (6.7).
4. Using (6.12) to estimate each term on the right hand side of (6.8), and recalling that
ρ = 1
2|y − y |, we conclude
|u(y) − u(y )| ≤ 2C
|y − y |
2
p−n
p
u W1,p(IRn). (6.13)
This shows that u is H¨older continuous with exponent γ = p−n
p .
5. To estimate supy |u(y)| we observe that, by (6.13), for some constant C1 one has
|u(y)| ≤ |u(x)| + C1 u W1,p(IRn) for all x ∈ B(y, 1) .
Taking the average of the right hand side over the ball centered at y with radius one we obtain
|u(y)| ≤ −
B(y,1)
|u(x)| dx + C1 u W1,p(IRn) ≤ C2 u Lp(IRn) + C1 u W1,p(IRn). (6.14)
28
6. Together, (6.13)-(6.14) yield
u C0,γ(IRn)
.
= sup
y
|u(y)| + sup
y=y
|u(y) − u(y )|
|y − y |γ
≤ C u W1,p(IRn) ,
for some constant C depending only on p and n.
Since C∞ is dense in W1,p, Morrey’s inequality yields
Corollary 6.1 (embedding). Let Ω ⊂ IRn be a bounded open set with C1 boundary. Assume
n < p < ∞ and set γ
.
= 1 − n
p > 0. Then every function f ∈ W1,p(Ω) coincides a.e. with a
function ˜f ∈ C0,γ(Ω). Moreover, there exists a constant C such that
˜f C0,γ ≤ C f W1,p for all f ∈ W1,p
(Ω). (6.15)
Proof. 1. Let Ω
.
= {x ∈ IRn ; d(x, Ω) < 1} be the open neighborhood of radius one around
the set Ω. By Theorem 4.3 there exists a bounded extension operator E, which extends each
function f ∈ W1,p(Ω) to a function Ef ∈ W1,p(IRn) with support contained inside Ω.
2. Since C1(IRn) is dense in W1,p(IRn), we can find a sequence of functions gn ∈ C1(IRn)
converging to Ef in W1,p(IRn). By Morrey’s inequality
lim sup
m,n→∞
gm − gn C0,γ(IRn) ≤ C lim sup
m,n→∞
gm − gn W1,p(IRn) = 0.
This proves that the sequence (gn)n≥1 is a Cauchy sequence also in the space C0,γ. Therefore
it converges to a limit function g ∈ C0,γ(IRn), uniformly for x ∈ IRn.
3. Since gn → Ef in W1,p(IRn), we also have g(x) = (Ef)(x) for a.e. x ∈ IRn. In particular,
g(x) = f(x) for a.e. x ∈ Ω. Since the extension operator E is bounded, from the bound (6.2)
we deduce (6.15).
6.2 The Gagliardo-Nirenberg inequality
Next, we study the case 1 ≤ p < n. We define the Sobolev conjugate of p as
p∗ .
=
np
n − p
> p. (6.16)
Notice that p∗ depends not only on p but also on the dimension n of the space. Indeed,
1
p∗
=
1
p
−
1
n
. (6.17)
As a preliminary, we describe a useful application of the generalized H¨older’s inequality. Let
n−1 non-negative functions g1, g2, . . . , gn−1 ∈ L1(Ω) be given. Since g
1
n−1
i ∈ Ln−1(Ω) for each
i, using the generalized H¨older inequality one obtains
Ω
g
1
n−1
1 g
1
n−1
2 · · · g
1
n−1
n−1 ds ≤
n−1
i=1
g
1
n−1
i Ln−1 =
n−1
i=1
gi
1
n−1
L1 . (6.18)
29
Theorem 6.2 (Gagliardo-Nirenberg inequality). Assume 1 ≤ p < n. Then there exists
a constant C, depending only on p and n, such that
f Lp∗
(IRn) ≤ C f Lp(IRn) for all f ∈ C1
c (IRn
) . (6.19)
3
(s ,x ,x )(x ,x ,x )1 2 3 1 2 3
x1
x 2
x
Figure 9: Proving the Gagliardo-Nirenberg inequality. The integral
∞
−∞
|Dx1
f(s1, x2, x3)| ds1 depends
on x2, x3 but not on x1. Similarly, the integral
∞
−∞
|Dx2
f(x1, s2, x3)| ds2 depends on x1, x3 but not
on x2.
Proof. 1. For each i ∈ {1, . . . , n} and every point x = (x1, . . . , xi, . . . , xn) ∈ IRn, since f has
compact support we can write
f(x1, . . . , xi, . . . , xn) =
xi
−∞
Dxi f(x1, . . . , si, . . . , xn) dsi .
In turn, this yields
|f(x1, . . . , xn)| ≤
∞
−∞
Dxi f(x1, . . . , si, . . . , xn) dsi 1 ≤ i ≤ n ,
|f(x)|
n
n−1 ≤
n
i=1
∞
−∞
|Dxi f(x1, . . . , si, . . . , xn)| dsi
1
n−1
. (6.20)
We now integrate (6.20) w.r.t. x1. Observe that the first factor on the right hand side does
not depend on x1. This factor behaves like a constant and can be taken out of the integral.
The product of the remaining n − 1 factors is handled using (6.18). This yields
∞
−∞
|f|
n
n−1 dx1 ≤
∞
−∞
|Dx1 f| ds1
1
n−1 ∞
−∞
n
i=2
∞
−∞
|Dxi f| dsi
1
n−1
dx1
≤
∞
−∞
|Dx1 f| ds1
1
n−1
n
i=2
∞
−∞
∞
−∞
|Dxi f| dsi dx1
1
n−1
.
(6.21)
Notice that the second inequality was obtained by applying the generalized H¨older inequality
to the n − 1 functions gi = ∞
−∞ |Dxi f|dsi, i = 2, . . . , n.
We now integrate both sides of (6.21) w.r.t. x2. Observe that one of the factors appearing in
the product on the right hand side of (6.21) does not depend on the variable x2 (namely, the
one involving integration w.r.t. s2). This factor behaves like a constant and can be taken out
30
of the integral. The product of the remaining n − 1 factors is again estimated using H¨older’s
inequality. This yields
∞
−∞
∞
−∞
|f|
n
n−1 dx1 dx2 ≤
∞
−∞
∞
−∞
|Dx1 f| dx1dx2
1
n−1 ∞
−∞
∞
−∞
|Dx2 f| dx1dx2
1
n−1
×
n
i=3
∞
−∞
∞
−∞
∞
−∞
|Dxi f| dsi dx1dx2
1
n−1
.
(6.22)
Proceeding in the same way, after n integrations we obtain
∞
−∞
· · ·
∞
−∞
|f|
n
n−1 dx1 · · · dxn ≤
n
i=1
∞
−∞
· · ·
∞
−∞
|Dxi f| dx1 · · · dxn
1
n−1
≤
IRn
| f| dx
n
n−1
.
(6.23)
This already implies
f Ln/(n−1) =
IRn
|f|
n
n−1 dx
n−1
n
≤
IRn
| f| dx , (6.24)
proving the theorem in the case where p = 1 and p∗ = n
n−1.
2. To cover the general case where 1 < p < n, we apply (6.24) to the function
g
.
= |f|β
with β
.
=
p(n − 1)
n − p
. (6.25)
Using the standard H¨older’s inequality one obtains
IRn
|f|
βn
n−1 dx
n−1
n
≤
IRn
β|f|β−1
| f| dx
≤ β
IRn
|f|
(β−1)p
p−1 dx
p−1
p
IRn
| f|p
dx
1
p
.
(6.26)
Our choice of β in (6.25) yields
(β − 1)p
p − 1
=
βn
n − 1
=
np
n − p
= p∗
.
Therefore, from (6.26) it follows
IRn
|f|p∗
dx
n−1
n
≤ β
IRn
|f|p∗
dx
p−1
p
IRn
| f|p
dx
1
p
.
Observing that n−1
n − p−1
p = n−p
np = 1
p∗ , we conclude
IRn
|f|p∗
dx
1
p∗
≤ C
IRn
| f|p
dx
1
p
.
31
If the domain Ω ⊂ IRn is bounded, then Lq(Ω) ⊆ Lp∗
(Ω) for every q ∈ [1, p∗]. Using the
Gagliardo-Nirenberg inequality we obtain
Corollary 6.2 (embedding). Let Ω ⊂ IRn be a bounded open domain with C1 boundary,
and assume 1 ≤ p < n. Then, for every q ∈ [1, p∗] with p∗ .
= np
n−p , there exists a constant C
such that
f Lq(Ω) ≤ C f W1,p(Ω) for all f ∈ W1,p
(Ω) . (6.27)
Proof. Let Ω
.
= {x ∈ IRn ; d(x, Ω) < 1} be the open neighborhood of radius one around the
set Ω. By Theorem 4.3 there exists a bounded extension operator E : W1,p(Ω) → W1,p(IRn),
with the property that Ef is supported inside Ω, for every f ∈ W1,p(Ω). Applying the
Gagliardo-Nirenberg inequality to Ef, for suitable constants C1, C2, C3 we obtain
f Lq(Ω) ≤ C1 f Lp∗
(Ω) ≤ C2 Ef Lp∗
(IRn) ≤ C3 f W1,p(Ω) .
6.3 High order Sobolev estimates
Let Ω ⊂ IRn be a bounded open set with C1 boundary, and let u ∈ Wk,p(Ω). The number
k −
n
p
will be called the net smoothness of u. As in Fig. 10, let m be the integer part and let
0 ≤ γ < 1 be the fractional part of this number, so that
k −
n
p
= m + γ . (6.28)
In the following, we say that a Banach space X is continuously embedded in a Banach
space Y if X ⊆ Y and there exists a constant C such that
u Y ≤ C u X for all u ∈ X .
k
0 1 2 m
m+ γ
nk −_
p
Figure 10: Computing the “net smoothness” of a function f ∈ Wk,p
⊂ Cm,γ
.
Theorem 6.3 (general Sobolev embeddings). Let Ω ⊂ IRn be a bounded open set with
C1 boundary, and consider the space Wk,p(Ω). Let m, γ be as in (6.28). Then the following
continuous embeddings hold.
32
(i) If k − n
p < 0 then Wk,p(Ω) ⊆ Lq(Ω), with 1
q = 1
p − k
n = 1
n
n
p − k .
(ii) If k − n
p = 0, then Wk,p(Ω) ⊆ Lq(Ω) for every 1 ≤ q < ∞.
(iii) If m ≥ 0 and γ > 0, then Wk,p(Ω) ⊆ Cm,γ(Ω).
(iv) If m ≥ 1 and γ = 0, then for every 0 ≤ γ < 1 one has Wk,p(Ω) ⊆ Cm−1,γ (Ω) .
Remark 6.1 Functions in a Sobolev space are only defined up to a set of measure zero. More
precisely, by saying that Wk,p(Ω) ⊆ Cm,γ(Ω) we mean the following. For every u ∈ Wk,p(Ω)
there exists a function ˜u ∈ Cm,γ(Ω) such that ˜u(x) = u(x) for a.e. x ∈ Ω. Moreover, there
exists a constant C, depending on k, p, m, γ but not on u, such that
u Cm,γ(Ω) ≤ C ˜u Wk,p(Ω) .
Proof of the theorem. 1. We start by proving (i). Assume k − n
p < 0 and let u ∈ Wk,p(Ω).
Since Dαu ∈ W1,p(Ω) for every |α| ≤ k − 1, the Gagliardo-Nirenberg inequality yields
Dα
u Lp∗
(Ω) ≤ C u Wk,p(Ω) |α| ≤ k − 1 .
Therefore u ∈ Wk−1,p∗
(Ω), where p∗ is the Sobolev conjugate of p.
This argument can be iterated. Set p1 = p∗, p2
.
= p∗
1, . . . , pj
.
= p∗
j−1. By (6.17) this means
1
p1
=
1
p
−
1
n
, . . .
1
pj
=
1
p
−
j
n
,
provided that jp < n. Using the Gagliardo-Nirenberg inequality several times, we obtain
Wk,p
(Ω) ⊆ Wk−1,p1
(Ω) ⊆ Wk−2,p2
(Ω) ⊆ · · · ⊆ Wk−j,pj
(Ω) . (6.29)
After k steps we find that u ∈ W0,pk (Ω) = Lpk (Ω), with 1
pk
= 1
p − k
n = 1
q . Hence pk = q and
(i) is proved.
2. In the special case kp = n, repeating the above argument, after k − 1 steps we find
1
pk−1
=
1
p
−
k − 1
n
=
1
n
.
Therefore pk−1 = n and
Wk,p
(Ω) ⊂ W1,n
(Ω) ⊆ W1,n−ε
(Ω)
for every ε > 0. Using the Gagliardo-Nirenberg inequality once again, we obtain
u ∈ W1,n−ε
(Ω) ⊆ Lq
(Ω) q =
n(n − ε)
n − (n − ε)
=
n2 − εn
ε
.
33
Since ε > 0 was arbitrary, this proves (ii).
3. To prove (iii), assume that m ≥ 0 and γ > 0 and let u ∈ Wk,p(Ω). We use the inequalities
(6.29), choosing j to be the smallest integer such that pj > n. We thus have
1
p
−
j
n
=
1
pj
<
1
n
<
1
p
−
j − 1
n
, u ∈ Wk−j,pj
(Ω) .
Hence, for every multi-index α with |α| ≤ k − j − 1, Morrey’s inequality yields
Dα
u ∈ W1,pj
(Ω) ⊆ C0,γ
(Ω) with γ = 1 −
n
pj
= 1 −
n
p
+ j .
Since α was any multi-index with length ≤ k − j − 1, the above implies
u ∈ Ck−j−1 ,γ
(Ω).
To conclude the proof of (iii), it suffices to check that
k −
n
p
= (k − j − 1) + 1 −
n
p
+ j ,
so that m = k − j − 1 is the integer part of the number k − n
p , while γ is its fractional part.
4. To prove (iv), assume that m ≥ 1 and j
.
= n
p is an integer. Let u ∈ Wk,p(Ω), and fix any
multi-index with |α| ≤ j −1. Using the Gagliardo-Nirenberg inequality as in step 2, we obtain
Dα
u ∈ Wk−j,p
(Ω) ⊆ W1,q
(Ω)
for every 1 < q < ∞. Hence, by Morrey’s inequality
Dα
u ∈ W1,q
(Ω) ⊆ C
0,1−n
q (Ω).
Since q can be chosen arbitrarily large, this proves (iv).
Example 9. Let Ω be the open unit ball in IR5, and assume u ∈ W4,2(Ω). Applying two
times the Nirenberg-Gagliardo inequality and then Morrey’s inequality, we obtain
u ∈ W4,2
(Ω) ⊂ W3, 10
3 (Ω) ⊂ W2,10
(Ω) ⊂ C1, 1
2 (Ω) .
Observe that the net smoothness of u is k − n
p = 4 − 5
2 = 1 + 1
2.
7 Compact embeddings
Let Ω ⊂ IRn be a bounded open set with C1 boundary. In this section we study the embedding
W1,p(Ω) ⊂ Lq(Ω) in greater detail and show that, when 1
q > 1
p − 1
n , this embedding is compact.
Namely, from any sequence (um)m≥1 which is bounded in W1,p one can extract a subsequence
which converges in Lq.
34
As a preliminary we observe that, if p > n, then every function u ∈ W1,p(Ω) is H¨older
continuous. In particular, if (um)m≥1 is a bounded sequence in W1,p(Ω) then the functions um
are equicontinuous and uniformly bounded. By Ascoli’s compactness theorem we can extract
a subsequence (umj )j≥1 which converges to a continuous function u uniformly on Ω. Since Ω
is bounded, this implies umj − u Lq(Ω) → 0 for every q ∈ [1, ∞]. This already shows that the
embedding W1,p(Ω) ⊂ Lq(Ω) is compact whenever p > n and 1 ≤ q ≤ ∞.
In the remainder of this section we thus focus on the case p < n. By the Gagliardo-Nirenberg
inequality, the space W1,p(Ω) is continuously embedded in Lp∗
(Ω), where p∗ = np
n−p . In turn,
since Ω is bounded, for every 1 ≤ q ≤ p∗ we have the continuous embedding Lp∗
(Ω) ⊆ Lq(Ω).
Theorem 7.1 (Rellich-Kondrachov compactness theorem). Let Ω ⊂ IRn be a bounded
open set with C1 boundary. Assume 1 ≤ p < n. Then for each 1 ≤ q < p∗ .
= np
n−p one has the
compact embedding
W1,p
(Ω) ⊂⊂ Lq
(Ω).
Proof. 1. Let (um)m≥1 be a bounded sequence in W1,p(Ω). Using Theorem 4.3 on the
extension of Sobolev functions, we can assume that all functions um are defined on the entire
space IRn and vanish outside a compact set K:
Supp(um) ⊆ K ⊂ Ω
.
= B(Ω, 1) . (7.1)
Here the right hand side denotes the open neighborhood of radius one around the set Ω.
Since q < p∗ and Ω is bounded, we have
um Lq(IRn) = um Lq(Ω)
≤ C um Lp∗
(Ω)
≤ C um W1,p(Ω)
for some constants C, C . Hence the sequence um is uniformly bounded in Lq(IRn).
2. Consider the mollified functions uε
m
.
= Jε ∗ um. By (7.1) we can assume that all these
functions are supported inside Ω. We claim that
uε
m − um Lq(Ω)
→ 0 as ε → 0, uniformly w.r.t. m. (7.2)
Indeed, if um is smooth, then (performing the changes of variable y = εy and z = x − εty)
uε
m(x) − um(x) =
|y |<ε
Jε(y ) [um(x − y ) − um(x)] dy
=
|y|<1
J(y)[um(x − εy) − um(x)] dy
=
|y|<1
J(y)
1
0
d
dt
um(x − εty) dt dy
= − ε
|y|<1
J(y)
1
0
um(x − εty) · y dt dy .
35
In turn, this yields
Ω
|uε
m(x) − um(x)| dx ≤ ε
Ω |y|≤1
J(y)
1
0
| um(x − εty)| dt dy dx
≤ ε
Ω
| um(z)| dz.
By approximating um in W1,p with a sequence of smooth functions, we see that the same
estimate remains valid for all functions um ∈ W1,p(Ω). We have thus shown that
uε
m − um L1(Ω)
≤ ε um L1(Ω)
≤ εC um W1,p(Ω)
, (7.3)
for some constant C. Since the norms um W1,p satisfy a uniform bound independent of m,
this already proves our claim (7.2) in the case q = 1.
3. To prove (7.2) also for 1 < q < p∗, we now use the interpolation inequality for Lp norms.
Choose 0 < θ < 1 such that
1
q
= θ · 1 + (1 − θ) ·
1
p∗
.
Then
uε
m − um Lq(Ω)
≤ uε
m − um
θ
L1(Ω)
· uε
m − um
1−θ
Lp∗
(Ω)
≤ C0 εθ
. (7.4)
for some constant C0 independent of m. Indeed, in the above expression, the L1 norm is
bounded by (7.3), while the Lp∗
norm is bounded by a constant, because of the GagliardoNirenberg
inequality.
4. Fix any δ > 0, and choose ε > 0 small enough so that (7.4) yields
uε
m − um Lq(Ω)
≤ C0 εθ
≤
δ
2
for all m ≥ 1 .
Recalling that uε
m = Jε ∗ um, we have
uε
m L∞ ≤ Jε L∞ um L1 ≤ C1,
uε
m L∞ ≤ Jε L∞ um L1 ≤ C2,
where C1, C2 are constants depending on ε but not on m. The above inequalities show that,
for each fixed ε > 0, the sequence (uε
m)m≥1 is uniformly bounded and equicontinuous. By
Ascoli’s compactness theorem, there exists a subsequence (uε
mj
) which converges uniformly on
Ω to some continuous function uε. We now have
lim sup
j,k→∞
umj − umk Lq
≤ lim sup
j,k→∞
umj − uε
mj Lq + uε
mj
− uε
Lq + uε
− uε
mk Lq + uε
mk
− umk Lq
≤ δ
2 + 0 + 0 + δ
2 .
(7.5)
36
5. The proof is now concluded by a standard diagonalization argument. By the previous step
we can find an infinite set of indices I1 ⊂ IN such that the subsequence (um)m∈I1 satifies
lim sup
,m→∞, ,m∈I1
u − um Lq ≤ 2−1
.
By induction on j = 1, 2, . . ., after Ij−1 has been constructed, we choose an infinite set of
indices Ij ⊂ Ij−1 ⊂ IN such that the subsequence (um)m∈Ij satisfies
lim sup
,m→∞, ,m∈Ij
u − um Lq ≤ 2−j
.
After the subsets Ij have been constructed for all j ≥ 1, again by induction on j we choose
a sequence of integers m1 < m2 < m3 < · · · such that mj ∈ Ij for every j. The subsequence
(umj )j≥1 satisfies
lim sup
j,k→∞
umj − umk Lq = 0 .
Therefore this subsequence is Cauchy, and converges to some limit u ∈ Lq.
As a first application of the compact embedding theorem, we now prove and estimate on the
difference between a function u and its average value on the domain Ω.
Theorem 7.2 (Poincare’s inequality - II). Let Ω ⊂ IRn be a bounded, connected open set
with C1 boundary, and let p ∈ [1, ∞]. Then there exists a constant C depending only on p and
Ω such that
u − −
Ω
u dx
Lp(Ω)
≤ C u Lp(Ω) , (7.6)
for every u ∈ W1,p(Ω) .
Proof. If the conclusion were false, one could find a sequence of functions uk ∈ W1,p(Ω) with
uk − −
Ω
uk dx
Lp(Ω)
> k uk Lp(Ω)
for every k = 1, 2, . . . Then the renormalized functions
vk
.
=
uk − −
Ω
uk dx
uk − −
Ω
uk dx
Lp(Ω)
satisfy
−
Ω
vk dx = 0 , vk Lp(Ω) = 1, Dvk Lp(Ω) <
1
k
k = 1, 2, . . . (7.7)
Since the sequence (vk)k≥1 is bounded in W1,p(Ω), if p < ∞ we can use the Rellich-Kondrachov
compactness theorem and find a subsequence that converges in Lp(Ω) to some function v. If
p > n, then by (6.15) the functions vk are uniformly bounded and H¨older continuous. Using
Ascoli’s compactness theorem we can thus find a subsequence that converges in L∞(Ω) to
some function v.
37
By (7.7), the sequence of weak gradients also converges, namely vk → 0 in Lp(Ω). By
Lemma 1.3, the zero function is the weak gradient of the limit function v.
We now have
−
Ω
v dx = lim
k→∞
−
Ω
vkdx = 0.
Moreover, since v = 0 ∈ Lp(Ω), By Corollary 2.1 the function v must be constant on the
connected set Ω, hence v(x) = 0 for a.e. x ∈ Ω. But this is in contradiction with
v Lp(Ω) = lim
k→∞
vk Lp(Ω) = 1.
8 Differentiability properties
By Morrey’s inequality, if Ω ⊂ IRn and w ∈ W1,p(Ω) with p > n, then w coincides a.e. with a
H¨older continuous function. Indeed, after a modification on a set of measure zero, we have
|w(x) − w(y)| ≤ C|x − y|
1−n
p
B(x, |y−x|)
| w(z)|p
dz
1/p
. (8.1)
This by itself does not imply that u should be differentiable in a classical sense. Indeed, there
exist H¨older continuous functions that are nowhere differentiable. However, for functions in a
Sobolev space a much stronger differentiability result holds.
Theorem 8.1 (almost everywhere differentiability). Let Ω ⊆ IRn and let u ∈ W1,p
loc (Ω)
for some p > n. Then u is differentiable at a.e. point x ∈ Ω, and its gradient equals its weak
gradient.
Proof. Let u ∈ W1,p
loc (Ω). Since the weak derivatives are in Lp
loc, the same is true of the weak
gradient u
.
= (Dx1 u , . . . , Dxn u). By the Lebesgue differentiation theorem for a.e. x ∈ Ω we
have
−
B(x,r)
| u(x) − u(z)|p
dz → 0 as r → 0 . (8.2)
Fix a point x for which (8.2) holds, and define
w(y)
.
= u(y) − u(x) − u(x) · (y − x). (8.3)
Observing that w ∈ W1,p
loc (Ω), we can apply the estimate (8.1) and obtain
|w(y) − w(x)| = |w(y)| = |u(y) − u(x) − u(x) · (y − x)|
≤ C |y − x|
1−n
p
B(x, |y−x|)
| u(x) − u(z)|p
dz
1/p
≤ C |y − x| −
B(x, |y−x|)
| u(x) − u(z)|p
dz
1/p
38
for suitable constants C, C . Therefore
|w(y) − w(x)|
|y − x|
→ 0 as |y − x| → 0 .
By the definition of w at (8.3), this means that u is differentiable at x in the strong sense,
and its gradient coincides with its weak gradient.
9 Problems
1. Determine which of the following functionals defines a distribution on Ω ⊆ IR.
(i) Λ(φ) =
∞
k=1
k! Dk
φ(k), with Ω = IR.
(ii) Λ(φ) =
∞
k=1
2−k
Dk
φ(1/k), with Ω = IR.
(iii) Λ(φ) =
∞
k=1
φ(1/k)
k
, with Ω = IR.
(iv) Λ(φ) =
∞
0
φ(x)
x2
dx, with Ω = ]0, ∞[ .
2. Give a direct proof that, if f ∈ W1,p( ]a, b[ ) for some a < b and 1 < p < ∞, then, by
possibly changing f on a set of measure zero, one has
|f(x) − f(y)| ≤ C |x − y|
1− 1
p for all x, y ∈ ]a, b[ .
Compute the best possible constant C.
3. Consider the open square
Q
.
= {(x1, x2) ; 0 < x1 < 1, 0 < x2 < 1} ⊂ IR2
.
Let f ∈ W1,1(Q) be a function whose weak derivative satisfies Dx1 f(x) = 0 for a.e. x ∈ Q.
Prove that there exists a function g ∈ L1([0, 1]) such that
f(x1, x2) = g(x2) for a.e. (x1, x2) ∈ Q .
4. Let Ω ⊆ IRn be an open set and assume f ∈ L1
loc(Ω). Let g = Dx1 f be the weak derivative
of f w.r.t. x1. If f is C1 restricted to an open subset Ω ⊆ Ω, prove that g coincides with the
partial derivative ∂f/∂x1 at a.e. point x ∈ Ω .
39
5. (i) Prove that, if u ∈ W1,∞(Ω) for some open, convex set Ω ⊆ IRn, then u coincides
a.e. with a Lipschitz continuous function.
(ii) Show that there exists a (non-convex), connected open set Ω ⊂ IRn and a function u ∈
W1,∞(Ω) that does not coincide a.e. with a Lipschitz continuous function.
6. Let Ω = B(0, 1) be the open unit ball in IRn, with n ≥ 2. Prove that the unbounded
function ϕ(x) = log log 1 + 1
|x| is in W1,n(Ω).
7. Let Ω = ]0, 1[. Consider the linear map T : C1([0, 1]) → IR defined by Tf = f(0).
Show that this map can be continuously extended, in a unique way, to a linear functional
T : W1,1(Ω) → IR.
8. Let Ω ⊂ IRn be a bounded open set, with smooth boundary. For every continuous function
f : Ω → IR, define the trace
T : C0
(Ω) → C0
(∂Ω)
by letting Tf be the restriction of f to the boundary ∂Ω. Show that the operator T cannot be
continuously extended as a map from Lp(Ω) into Lp(Ω), for any 1 ≤ p < ∞. In other words,
a generic function f ∈ Lp(Ω) does not have trace on the boundary ∂Ω.
9. To construct the trace of a function u : W1,p(Ω), consider the following approach.
(i) Using Theorem 4.3, extend u to a function Eu ∈ W1,p(IRn) defined on the entire space
IRn, including the boundary of Ω.
(ii) Define the trace of u as the restriction of Eu to the boundary of Ω, namely Tu
.
= Eu|∂Ω.
Explain why this approach is fundamentally flawed.
10. Let V ⊂ IRn be a subspace of dimension m and let V ⊥ be the perpendicular subspace, of
dimension n − m. Let u ∈ W1,p(IRn) with m < p < n. Show that, after a modification on a
set of measure zero, the following holds.
(i) For a.e. y ∈ V ⊥ (w.r.t. the n−m dimensional measure), the restriction of u to the affine
subspace y + V is H¨older continuous with exponent γ = 1 − m
p .
(ii) The pointwise value u(y) is well defined for a.e. y ∈ V ⊥. Moreover
V ⊥
|u(y)| dy ≤ C · u W1,p(IRn)
for some constant C depending on m, n, p but not on u.
40
11. Determine for which values of p ≥ 1 a generic function f ∈ W1,p(IR3) admits a trace
along the x1-axis. In other words, set Γ
.
= {(t, 0, 0) ; t ∈ IR} ⊂ IR3 and consider the map
T : C1
c (IR3) → Lp(Γ), where Tf = f|Γ is the restriction of f to Γ. Find values of p such that
this map admits a continuous extension T : W1,p(IR3) → Lp(Γ).
12. When k = 0, by definition W0,p(Ω) = Lp(Ω). If 1 ≤ p < ∞ prove that W0,p
0 (Ω) = Lp(Ω)
as well. What is W0,∞
0 (Ω) ?
13. Let ϕ : IR → [0, 1] be a smooth function such that
ϕ(r) =
1 if r ≤ 0 ,
0 if r ≥ 1 .
Given any f ∈ Wk,p(IRn), prove that the functions fk(x)
.
= f(x) ϕ(|x| − k) converge to f
in Wk,p(IRn), for every k ≥ 0 and 1 ≤ p < ∞. As a consequence, show that Wk,p
0 (IRn) =
Wk,p(IRn).
14. Let IR+
.
= {x ∈ IR ; x > 0} and assume u ∈ W2,p(IR+). Define the symmetric extension
of u by setting Eu(x)
.
= u(|x|). Prove that Eu ∈ W1,p(IR) but Eu /∈ W2,p(IR), in general.
15. Let u ∈ C1
c (IRn) and fix p, q ∈ [1, ∞[ . For a given λ > 0, consider the rescaled function
uλ(x)
.
= u(λx).
(i) Show that there exists an exponent α, depending on n, p, such that
uλ Lp(IRn) = λα
u Lp(IRn) .
(ii) Show that there exists an exponent β, depending on n, q, such that
uλ Lq(IRn) = λβ
u Lq(IRn) .
(iii) Determine for which values of n, p, q one has α = β. Compare with (6.16).
16. Let Ω ⊆ IRn be a bounded open set, with C1 boundary. Let (um)m≥1 be a sequence of
functions which are uniformly bounded in H1(Ω). Assuming that um −u L2 → 0, prove that
u ∈ H1(Ω) and
u H1 ≤ lim inf
m→∞
um H1 .
17. Let Ω
.
= {(x1, x2) ; x2
1 + x2
2 < 1} be the open unit disc in IR2, and let Ω0
.
= Ω \ {(0, 0)}
be the unit disc minus the origin. Consider the function f(x)
.
= 1 − |x|. Prove that



f ∈ W1,p
0 (Ω) for 1 ≤ p < ∞ ,
f ∈ W1,p
0 (Ω0) for 1 ≤ p ≤ 2 ,
f /∈ W1,p
0 (Ω0) for 2 < p ≤ ∞ .
41
g
x0
1_
n
_1
1
nf
f
0 1
f
n
Figure 11: Left: the function f can be approximated in W1,p
with functions fn having compact
support in Ω. Right: the function f can be approximated in W1,2
with functions gn having compact
support in Ω0.
18. Let Ω = {(x1, x2) ; 0 < x1 < 1, 0 < x2 < 1} be the open unit square in IR2.
(i) If u ∈ H1(Ω) satisfies
meas {x ∈ Ω ; u(x) = 0} > 0 , u(x) = 0 for a.e. x ∈ Ω
prove that u(x) = 0 for a.e. x ∈ Ω.
(ii) For every α > 0, prove that there exists a constant Cα with the following property. If
u ∈ H1(Ω) is a function such that meas({x ∈ Ω ; u(x) = 0}) ≥ α, then
u L2(Ω) ≤ Cα u L2(Ω) . (9.4)
19. Let (un)n≥1 be a sequence of functions in the Hilbert space H2(IR3)
.
= W2,2(IR3). Assume
that
lim
n→∞
un(x) = u(x) for all x ∈ IR3
, un H2 ≤ M for all n.
Prove that the limit function u coincides a.e. with a continuous function.
20.
(i) Find two functions f, g ∈ L1
loc(IRn) such that the product f · g is not locally integrable.
(ii) Show that, if f, g ∈ L1
loc(IR) are both weakly differentiable, then the product f · g is also
weakly differentiable and satisfies the usual product rule: Dx(fg) = (Dxf) · g + f · (Dxg).
(iii) Find two functions f, g ∈ L1
loc(IRn) (with n ≥ 2) with the following properties. For every
i = 1, . . . , n the first order weak derivatives Dxi f, Dxi g are well defined. However, the product
f · g does not have any weak derivative (on the entire space IRn).
42